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O'BRIEN'S

MATHEMATICAL TRACTS,

PART I.

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MATHEMATICAL TRACTS,

PART I.

MATHEMATICAL TRACTS,

PART I.

LAPLACE'S COEFFICIENTS, THE FIGURE OF THE EARTH,

THE MOTION OF A RIGID BODY ABOUT ITS CENTER OF GRAVITY,

AND

PRECESSION AND NUTATION.

MATTHEW O'BRIEN, B.A.,

•MATHEMATICAL LECTURER OP CAIUS COLLEGE.

CAMBRIDGE:

PRINTED AT THE UNIVERSITY PRESS,

FOR J. & J. J. DEIGHTON, TRINITY STREET;

AND

JOHN W. PARKER, LONDON.

M.DCCC.XL.

Stack Annex

233

PREFACE.

THE subjects treated of in the following Tracts are, Laplace's Coefficients; the Investigation of the Figure of the Earth on the Hypothesis of its Original Fluidity ; the Equations of Motion of a Rigid Body about its Center of Gravity; and the Application of these Equations to the case of the Earth. The first of these subjects should be familiar to every Mathematical Student, both for its own sake, and also on account of the many branches of Physical Science to which it is applicable. The second sub- ject is extremely interesting as a physical theory, bearing upon the original state of the Earth and of the planetary bodies; it is also well worthy of attention on account of the important and exten- sive observations which have been made in order to verify it. The Author has put both these sub- jects together, commencing with the Figure of the Earth, and introducing Laplace's Coefficients when occasion required them; this being perhaps the best

VI PREFACE.

and simplest way of exhibiting the nature and use of these coefficients.

The Author has treated some parts of these sub- jects differently from the manner in which they are usually treated, and he hopes that by so doing he has avoided some intricate reasoning and trou- blesome calculation, and made the whole more accessible to students of moderate mathematical at- tainments than it has hitherto been.

In calculating the attractions of the Earth on any particle, he has arrived at the correct results, without considering diverging series as inadmissible ; and this he conceives to be important, because there is evidently no good reason why a diverging series should not be as good a symbolical representative of a quantity as a converging series ; or why there should be any occasion to enquire whether a series is di- verging or converging, as long as we do not want to calculate its arithmetical value or determine its sign. Instances, it is true, have been brought for- wprd by Poisson* in which the use of diverging series appears to lead to error; but if the reason- ing employed in Chapter in. of these Tracts be not incorrect, this error is due to quite a different cause ;

• See Bowditch's Laplace. Vol. H. p. 167.

PREFACE. Til

as will be immediately perceived on referring to Ar- ticles 33, 34, 35, and 37.

The Author has deduced the equations of motion of a rigid body about its center of gravity by a method which he hopes will be found, less objec- tionable than that in which the composition and re- solution of angular velocities are employed, and less complex than that given by Laplace and Poisson ; he has also endeavoured to simplify the application of these equations to the case of the Earth.

In the First Part of these Tracts he has confined himself to the most prominent and important parts of each subject. In the Second Part, which will shortly be published, he intends, among other- things, to give some account of the controversies which Laplace's Co- efficients have given rise to ; to investigate more fully the nature and properties of these functions ; to give instances of their use in various problems ; for this purpose to explain the mathematical theory of Elec- tricity ; to consider more particularly the Equations of motion of a rigid body about its center of gra- vity, and the conclusions that may be drawn from them ; to give the theory of Jupiter's Satellites, and of Librations of the Moon ; 'and to say something on the subject of Tides.

Vlll PREFACE.

The Author has not given the investigation of the effect of the Earth's Oblateness on the motions of the Moon, but he has endeavoured to prove that this effect does not afford any additional evidence of the Earth's original Fluidity beyond that which may be obtained from the Figure of the Earth, and Law of Gravity.

MATHEMATICAL TRACTS, PART I.

CHAPTER I.

FIGURE OF THE EAUTH.

1. IT has been well ascertained, by extensive and accurate geodetical measurements, that the general figure of the Earth is that of an oblate surface of revolution, de- scribed about the axis of diurnal rotation : and this fact suggests the idea, that the diurnal rotation may be in some way or other the cause of this peculiar figure, especially if we consider that the Sun and planets, which all rotate like the Earth, appear also to have the same sort of oblate form of revolution about their axes of rotation.

The most obvious and natural way of accounting for the influence thus apparently exerted on the figures of the planetary bodies by their rotation, is to suppose that they may once have been in a state of fluidity ; for, con- ceive a fluid gravitating mass to be gradually put into a state of rotation round a fixed axis : it is evident that be- fore the motion commenced it would, according to a well- known hydrostatical law, be arranged all through in con- centric spherical strata of equal density ; but on the motion of rotation commencing a centrifugal force would arise, which would be greater at greater distances from the axis, and would therefore evidently produce an oblateness in the forms of the strata, leaving them still symmetrical with respect to the axis. Thus the hypothesis of the original fluidity of the bodies of our system, considered in connection with their rotation, accounts for their oblate form. 1

2

2. To account for the present solidity of the surface of our own planet, we may suppose that its temperature was originally so great as to keep it in a state of fusion, and that this was the cause of its fluidity ; but that, in the course of ages, it, at least its surface, has cooled down and hardened into its present consistence. This supposition is borne out by geological facts ; and it is by no means un- likely, if we consider that the principal body of our system is at present most probably in a state of fusion.

3. This hypothesis of the Earth's original fluidity re- ceives much confirmation from observations on the intensity and direction of the force of gravity ; for it follows from the hydrostatical law already alluded to, that the Earth, if fluid, ought to consist entirely of equidense strata of the same sort of form as the exterior surface*, and therefore the whole mass ought to be arranged symmetrically with respect to the axis of rotation, and nearly so with respect to the centre of that axis. Hence, the force of gravity, which is the resultant of the Earth's attraction and the centrifugal force, ought to be the same at all places in the same latitude, and nearly the same at all places in the same meridian.

Moreover it follows from another hydrostatical law, that the direction of this force of gravity ought to be every where perpendicular to the surface.

Now all this has been proved to be the case by nume- rous observations with pendulums, plumb-lines, levels, &c. (omitting very small variations, which may be easily ac- counted for in most cases). Hence, the hypothesis of the Earth's original fluidity is confirmed by the observations which have been made on the force of gravity.

4. But this hypothesis has been advanced almost to a moral certainty, by investigating precisely what effect it ought to have, if true, on the arrangement of the Earth's

* We suppose the Earth to be heterogeneous, because the pressure of the superincumbent mass must condense the central parts more than the superh'cial ; besides, the well-known fact of the mean density of the whole Earth being greater than the density of the superficial parts, proves that the Earth is not homo- geneous.

mass, and by comparing the result with observation; for it is found that if the hypothesis be true, the strata which com- pose the Earth ought to have not only an oblate form, but one very peculiar kind of oblate form ; and it is found that this result admits of most satisfactory comparison with ac- curate and varied observation, and actually coincides with it in a most remarkable manner; from which we may con- clude, almost with certainty, that the hypothesis is correct ; for it is extremely difficult to account in any other way for so marked an agreement with observation of such a very peculiar result.

5. The object of the following pages is to give an account of this interesting investigation, and to state briefly the manner in which its result may be tested by observation. In the first place, we shall determine the law of arrange- ment of the Earth's mass, on the hypothesis of its original fluidity, by means of Laplace's powerful and beautiful Ana- lysis; and in the next place, we shall deduce such results as shall admit of immediate comparison with observation. The most important of these results are ; The expression for the length of a meridian arc corresponding to a given difference of latitude, and. The law of variation of the force of gravity at different points of the Earths surface.

The other results which we shall deduce depend on certain assumptions respecting the law of density of the Earth, and are therefore not so important. We now proceed, in the first place, to determine the law of arrangement of the Earth's mass, as follows.

6. A heterogeneous jluid mass composed of par- ticles which attract each other inversely as the square of the distance rotates uniformly in relative equilibrium* round a fixed axis : to determine the law of its arrange- ment.

Take the axis of rotation as that of #, and let xyz be the co-ordinates of any particle $m, XYZ the resolved

* By relative equilibrium we mean that the particles of the mass, though actually moving, are at rest relatively to each other.

attractions ol' the mass on $w, p and p the density and pressure at the point (#ysf), and o> the angular velocity of the mass. Then, by the principles of Hydrostatics, we have

dp = p \Xdx + Ydy + Zdz + w8 (xdx + ydy)} ... (l) To calculate the expression (Xdcc + Ydy + Zdz), let $m be any attracting particle, and x'y'% its co-ordinates ; then we have

and similar expressions for Y and Z.

Now assume (F) to denote the expression

VV- *)2 + (y- y?+ (*' - *)2 '

i. e. the sum of each particle divided by its distance from §m. Then it is evident that

y dV dV dV

= Tx* ~~dy* ~dx'

and Xdfc + Ydy + Zdz = —— dx + —:—dy -\ -- dx. dx dy ' dss

and therefore the equation (1) becomes

The coefficient of p here is a complete differential ; hence by the principles of Hydrostatics, the necessary and sufficient conditions of equilibrium are, that the whole mass be arranged in strata of equal density, the general equation to any one of them being

C being a constant different for different strata, the exterior surface being one of these strata, since it is a free surface.

7- Hence the equations from which the problem is to be solved are

(A).

=

8. These equations are unfortunately very much in- volved in each other, so much so as to be scarcely manage- able ; for V must be found by integration between limits which depend on the form of the exterior stratum, and therefore on the equation (A) ; and also the law of density, and therefore the form of the internal strata, and therefore the equation (A), must be known in order to calculate F. But V is itself involved in (A), hence (A) cannot be made use of in calculating V. It will therefore be neces- sary to devise some way of eliminating F, without knowing what function it is. To do this in the general case is be- yond the present powers of analysis ; but in the particular case we are concerned with, the fact of the strata being all nearly spherical, introduces considerable simplification, and by using the ingenious analysis due to Laplace, we shall be able to eliminate V with comparative ease, at least, ap- proximately, but with quite sufficient accuracy.

9. In the first place, the strata being nearly spherical, we shall find it convenient to make use of polar instead of rectangular co-ordinates, and we shall accordingly transform our equations as follows :

Let r, 0, (f>, /, 0', 0', be the co-ordinates of $m and $m' respectively; r, 0, 0, signifying the same as in Hymers1 Geometry of three dimensions, page (77). Then we have

off = r sin 9 cos d),

y = r sin 0 sin <^>,

% = r cos 9, and similar expressions for v't y', %'.

Hence the equation (A) becomes C = V + — r- sin2 9,

2

and the equation (/?) becomes, observing that $m = p r'2 sin 9' dr d& dfi

c^ r

Jo Jo

\Xr'-2rr'|cos#cos#'+sin0sin0'cos(0-^)')}-fr'2

0 and r, being the limits of r', 0 and TT of 9', and 0 and 2?r of 0', 7*1 being the value of r at the surface, and there- fore in general a function of 0' and <p'. It will of course on this account be necessary to integrate first with respect to /, but it is no matter in what order we perform the integrations relative to 0' and <p', since their limits are constants.

If these integrations could be performed, V would come out a function of r, 0, and <^>, and unknown constants de- pending on the form of the strata and the law of density.

10. We shall find it convenient to put /* and /u? for cos 6 and cos 9' respectively, this will give

sin 9'd9' = - dp.',

and the limits of /u. will be — 1 and 1 ; or we may put dfi instead of — d//, if at the same time we reverse the limits of fji. Hence our equations become

(A'),

When for brevity we have put

cos 9 cos ff + sin 9 sin 9' cos (0 - <^>') = p, i. c. fifj! + \/l -fj? . \A - //* • cos (^> ~ 0') = P-

11. We shall now introduce into these equations the condition that the strata are nearly spherical. If the strata were actually spherical, the whole mass would be symmetri- cal with respect to the centre, and therefore V being the sum of each particle divided by its distance from $m would depend simply on the distance of $m from the centre, and therefore be the same at all points of the same stratum. We may hence conclude, that if the strata instead of being actually spherical be only nearly so, V also, though not actually the same, will yet be nearly the same at all points of the same stratum. Now the value of V for any stratum is given by the equation {A') i. e.

but V (as we have shewn) ought to be nearly constant at all points of this stratum, hence the variable part of it, viz.

— r2(l — /u2) must be always small : therefore since r2(l — fj?)

is not always small, wz must be so. We shall accordingly take ft>2 as the standard small quantity in our approxima- tions, neglecting its square and higher powers in the first approximation.

12. Now to8 being a small quantity, we may suppose the equation to any nearly spherical surface, and therefore to any of the strata, to be put in the form

where a is the radius of any sphere which nearly coincides with the stratum (that sphere, suppose, which includes the same volume as the stratum*), and aa?u is the small quan- tity to be added to a to make it equal to r, and therefore u in general will be some function of /u. and 0. Moreover, u will be a function of a also, otherwise the strata would be all similar surfaces, which of course we have no right to assume them to be ; a may be considered as the variable

* We make this supposition, at present,"for the sake of giving a definite idea of what a is ; hereafter it will be found an advantageous supposition.

8

parameter of the system of surfaces which the strata con- stitute. We shall introduce the variable a into our' equa- tions instead of r, and get every thing in terms of a, /x and <£, instead of r, /u, and 0; the advantages of this change will soon be perceived.

13. First, then, in the equation (^'), putting a (l+a>2w) instead, of r, and neglecting the squares and higher powers of <*r, we find

Next we shall make a similar substitution in the equa- tion (B') by putting r' = a (l + o>2w'), when u denotes what u becomes when a, /*, and 0 are exchanged for a', /, and <p' respectively.

Assume for brevity,

then in the equation (B') we shall have

* = /oai p/ (r, /) . da',

where Oj is the parameter of the exterior surface.

and substituting for r,f(r,r')=tf{r,(a'+a'u?u)}

dr 9 d(a'u)

and hence, since — 7 = 1 + to . — - — H^ , we have da da

9

Hence, neglecting the squares and higher powers of ft)2, the equation (B7) becomes

14. This expression for V is much more manageable than that in the equation (B1) ; for in the first place the limits are all constant, and therefore we may take them in any order we please; and in the second place, p, instead of depending on all the variables, as of course it did before, is now a function of one variable only, viz. a \ for each stratum being equidense throughout, it is evident that p is the same at every point of the same stratum, and therefore varies only when we pass from stratum to stratum, i. e. it varies with a alone.

Hence, changing the order of the integrations, and bringing p outside the integral signs relative to ^ and 0', we have

~ *r a'u' d* <*' da''

15. We have thus introduced into our equations the condition of the strata being nearly spherical.

We shall find it convenient to make a farther substitu- tion in this equation, viz. by putting r—a(l + <o2w), which, neglecting the squares and higher powers of <o2, gives

r P {r V-! /(a' a/> dfi' w

— ^ f /(a, a') dp.' d<b' aaJ<\ J -\"

'an (a' ir f f(a* a'^u> dfjL d<p} \ da''

10

16. The next thing we shall do is to perform the in- tegrations relative to /*' and 0' ; to do this we shall expand the quantity^y^aa'), which, since it represents

a'z \/a/2-2aa'p+a2'

a a

may be expanded in a series of powers either of — or — .

a a

The coefficients will evidently be the same whether we ex- pand it in powers of —. or of — , they will in fact be the a a

coefficients of the powers of h in the expansion of the quantity

1

We shall assume Qy, Qn Q2, &c. to denote these co- efficients, i. e. we shall assume

Q0 will evidently be unity. The rest of these coefficients will be rational and integral functions of p, i.e. of

fifi + V 1 — yU2 . V 1 — //2 . COS (0 — 0').

It is evident that they all become unity when p becomes unity ; for then

becomes , or 1 + h + h2 + &c.

We shall have no occasion however to determine their forms. They (and other functions of the same character) are the celebrated coefficients of Laplace ; they possess very remarkable properties, which we shall now digress to inves- tigate, as they wonderfully facilitate the integrations we have to perform, and enable us to eliminate V from the equations (A") and (5"'), with great facility, without know- ing its form.

CHAPTER II.

LAPLACE S COEFFICIENTS.

17. IN order to investigate the properties of the functions Q,, Qn Q2, SEC. introduced in the last chapter, we shall recur to the expression from which they were origin- ally derived, viz.

1

We meet with this expression constantly in physical problems, especially those in which attractions are con- cerned, and it is therefore worthy of particular consi- deration.

Assuming R to denote this expression, we have

R

'- x? + (yf- yf + (x'-)*'

and differentiating this equation twice relatively to xyx respectively,

dR __ (x-.v) dx ~ K*'-*)2+(y'-y)2 + (*'-*)2}* -JP (*'-*)

« .JP'V-.)-*

dx* dx

and similarly

fp Tt —

12

\

hence evidently

d-JR d*R d*R _ 1

* =0 (1).

18. We shall express this differential equation in terms of the polar co-ordinates r0(p instead of xyx. To facilitate the transformation we shall assume an auxiliary quantity s, such that

s = r sin 0 ;

and therefore since x = r sin 0 cos 0, and y = r sin 6 sin 0, we shall have

X = S COS 0,

y = s sin 0.

Then considering s and 0 as independent variables instead of x and y, we have dR _ dR dx dR dy ds dx ds dy ds

dR dR .

= — — cos 0 + — — sm0 (2).

dx dy

dzR d*R dzR d?R

and — — = — — cos^0+2 • — — cos 0 sin 0+ — — sm20...(3). ds* dx2 dxdy dy2

. dR dR dR

and -**,= - — * sin d> H s cosd> (4).

(fm dx dy

d*R d*R . d*R d?R

-r = -T-r*f Sin 0 — 2 — — . S^COS 0 Sin 0 H S2 COSZ(b

d0 dx2 dxdy dy2

dR ' dR

Equations (3) and (5) give

d?R l d*R d?R tfR 1 (dR dR

~TT + 1 T3T = T~T + ~T~9 --- — cos 0 + — — sin 0 ds* s2 d02 dx2 dy2 s \dx dy

d*R d?R 1 dR

by equation (2).

13

Now we have

x = r cos 9 s — r sin 0*,

and these equations connect % s r 9 in exactly the same way in which xys(f> are connected by the equations

X = S COS 0

y = s sin 0. Hence we may prove exactly as before, that

dR l (PR _ d'R d*R _ i dR d^Jr^i~d¥= "d**" + rf*2 ~ r~dr'" adding this to the equation (6), we have by (l), I. tfR rffl l^ d~R l dR l dB_ ? d0* +~d7+ r2 drF" « ds r~d7" Now by (2) and (4),

dR dR cos0 dR

— sin 0 + — —2- = — - , ds d(f) s dy

and hence observing as before, that x s r 9 are connected together in exactly the same way as x y s 0, we have dR . dR cos 9 dR

hence, substituting this value of — — in equation (7), and

ds

putting r sin 9 for s, and multiplying by r2, we have d*Jl cos 9 d_R_ I d^R. ,d*R dR_ _ J¥ + sin 9 ~d9 + shT^ dtf + r ~d7 + 'T ~dr ~ °' l d dR\ l d*R d f dR

which is the equation (l) expressed in polar co-ordinates.

* The author finds that he has been anticipated in making this use of the auxiliary quantity s, by the Cambridge Mathematical Journal.

14

19. Now K expressed in polar co-ordinates becomes 1

Vr -

which = — r

/ r r*

V 1 - %P - + -72

Qn n + &C.

(See Art. 16). Hence substituting this value of R in the equation just obtained, and putting the coefficient of r" equal to zero, since r is indeterminate, we find imme- diately

sin 0 dO \ d9 ] sin* 0 d(f>

which is a partial differential equation of the second order, connecting Qn with /a. and 0 ; of course, being such, it admits of an infinite number of solutions besides Qn. We shall have no occasion to solve it, but "we shall find it of use in investigating the properties of Qn. All solutions of it which are rational and integral functions of cos 0, sin 0, cos <f), sin <p (i. e. of /x, \/l-fj?9 cos (f), sin 0), are called Laplace's coefficients of the nth order, having been first brought into notice by Laplace in his Mecanique Celeste, Liv. in.: the equation itself may be called Laplace's equation. Why we restrict Laplace's coefficients to be rational and integral functions of /u> 'X/l - M8> cos 0 a°d sin (f>, will appear presently.

20. We may remark here that in consequence of the linearity of Laplace's equation, the sum of any number of

15

Laplace's coefficients of the wth order is also a Laplace's co- efficient of the wth order.

Also any constant quantity is a Laplace's coefficient of the order 0, for if F0 be a Laplace's coefficient of the order 0, we have

d

which equation is evidently satisfied by F0 = any constant,

and hence any constant is a Laplace's coefficient of the order 0.

It may easily be seen by trial that

a/i, and a \/l - /u2 cos (0 + /3)

are Laplace's coefficients of the order 1, a and /3 being constants ; and

a (1 - M2)> a jtA \A ~ M2 cos (0 + /3), a (l - fj?) cos (2 0 + /3),

are Laplace's coefficient of the order 2, and so on. We shall not have any occasion at present to determine the general expression for a Laplace's coefficient of the wth order, but, to give clear ideas, we shall just state that it may be put in the form

A0 Mn+4} (I -fjfyM^costy + aj -I- 4,(1 -ffiMn_ 2cos(2<p + a2) , &c ............. +A(l-

When AQ Al SEC. ... a^ a2 ... &c. are any constants, Mn Mn_^ &c. contain rational and integral functions of /u, of the di- mensions w, n — l, n -*2, &c. respectively*.

21. The first property we shall prove of Laplace's coefficients is this : If Ym and Zn be any two Laplace's coefficients of the mth and wth orders respectively, then

i I Ym Zndju. d(f) = 0, except when m = n.

Jo J -i

* We shall recur to this subject in Part n. of these Tracts.

16 For since ZM satisfies Laplace's equation, we have

n (n + 1) *" Ym Zn

Now integrating by parts, and observing that 1 - M2 = 0, at each limit we have

and similarly,

f"v fz-j r~ dr.dz J, Y"^d* = - J. Tp-jf**.

1 /^

* observing that Ym -•--* is the same at each limit, be- ct(p

cause Ym and Zn are functions of sin 0 and cos 0, and not of 0 simply ; hence substituting in (l)

n.(n + l) r*r

J0 J-\

r^ ri (/ 2,dr»

-I. L(i-^^'

In exactly the same way we may shew that

f'

-I.

* This is the reason why we have assumed Laplace's coefficients to be functions of sin <f> and cos <7>, and not of <j> simply.

17

Hence subtracting \n(n + 1) -m(m+ 1) } * Ym Zndnd<t>= 0 .

Now the factor n(n + l) — m(m + 1) does not = 0, except when m = n ; hence the other factor must be zero, hence

r* r Ymznd/uid(f>=o,

J0 J -} except when m = n.

22. Since Q0 = l (see Art. 16), we have

f

J —

= 0, when n is greater than 0,

= r^ f

'0 = 47T.

It need scarcely be remarked that Q0, Q15 Q2, &c. possess exactly the same properties with respect to p and 0' that they do with respect to /j. and (p.

23. We shall now have occasion to introduce a remarkable discontinuous function, but before we do so we shall give a simple example of functions of this description, in order to render our reasoning more satis- factory to those who have not been accustomed to them.

We may easily prove, by the aid of the exponential value of the cosine of an arc, that

*(.-f)

cos a + cos (a + /3) + cos (a + 2/3) ad inf. ... = — ;

suppose here that a = — , then we find

0 cos a + cos 3 a + cos 5 a + ... = -

2 sin a

18

hence this series is always zero, except when a= any mul- tiple of TT, in which case sin a becomes zero, and therefore

the series becomes - ; thus though each term of this series

varies continuously with a, the series itself varies discon- tinuously, being constantly zero, except when a passes through any of the values 0, ± TT, ± 2 ?r, &c.. when the series

suddenly becomes -; i.e., some unknown or indeterminate

quantity. To explain the nature of this series more clearly, we observe that

sin — | cos a 4- cos (a + /5) + SEC. } = - ^ sin f a ) ,

whatever be the values of a and /3 ; suppose a = — ;

then sin I a 1 becomes zero, whatever be the value of

a ; hence

sin a {cos a + cos 3a + cos 5a + &c. } = 0, for all values of a.

Now as long as a is not a multiple of TT, sin a will not be zero, and therefore

cos a + cos 3 a + ... &c.

must be zero ; but if a be any multiple of TT, then sin a will be zero, and our equation will be satisfied quite independent- ly of its other factor, and hence will give us no information as to the value of that factor; hence when a is any multiple of T, cos a + cos 3 a + &c. is some unknown or indeterminate quantity.

It is important to remark that the change in the value of cos a + cos 3a + &c., when a becomes a multiple of -a-, is perfectly sudden, for since the second member of the equa- tion is always absolutely zero, it is evident that as long as sin a is not actually zero, though it differs from it by ever so

19

small a quantity, cos a + cos 3 a + &c ---- must be so; for this reason cos a + cos 3 a + Sic. is called a discontinuous function.

24. We shall now bring forward the remarkable dis- continuous function we alluded to: it is the following series, viz.

Qo+ 3Q/+ &c- + (2w + 1) Q«+ &c. ad inf. this series is of exactly the same nature as that we have just considered, being a function of the variables /x and <£», which is always zero, except for certain particular values of these variables.

To shew this, we have

Q + Q,A + ... Qnh» + &c. = , ~ ,

VI - 2ph + h?

and differentiating this relatively to /*and multiplying by 2 h,

and adding these equations,

«,+ »(t*...(g. + .) «.*•+ &c. - (1--^¥

now here put h — 1 and we find

0

O, + 3 Q + &c. = - — — = , (2 - 2p)f

hence QQ+ 3Q7 + &c. = 0 for all values of />, except p

0

when it becomes - : now 0

P=fJLfl' + \ -/(Z2 ^l - /li'2 COS ( - '),

but cos (0 - <£') is not greater than 1 ;

20

hence,

1 -/*/*' or, squaring and reducing,

(fj. — p)* is not > 0,

which cannot be unless /u. = //, and this will give cos (<p - <£') = 1 ;

and therefore 0 - 0' is zero, or some multiple of STT; hence the series Q0 + 3 Qx + &c. is a discontinuous function, being always zero, except when M = M' and <f> ~ <p' = 2wrr, (m

being any integer,) in which case it becomes -. It is evi-

dent, as in the former case, that this series is perfectly discontinuous, being zero for all values of p that differ even in the least degree from unity, and then when p = 1, sud-

denly assuming the form - .

25. Now, wherever we have occasion to use the series Q0 + 3Ql + Sec., it will occur under integral signs relative to fi and 0, and the limits of <f) will be 0 and STT; hence, by what we have proved in the note* respecting the limits of

* If x be any arbitrary quantity occurring in any investigation, its differen- tial dx may be defined to be any small increment of x, made use of with the understanding that it is to be put equal to zero at the end of the investigation ; and if f(x) be any function of at, its differential df(x) may be defined to be the corresponding increment of /(#), that is,

The symbol j written before a differential, is generally taken to denote the quantity from which the differential is derived, that is,

/«*/(*) =/(*),

and the notation /**<*/(*) is taken to denote the difference f(x3)-f(xl): but this notation has a much more important signification : for in the equation

put for x successively the values .r,, Xi + dn, z1 + 2dn, &c., #, + (»- \)dx, and add the results, and we find

! + dx) + d/(-r, + 2dx) &c. + df{Xl + (n - 1) dx\

21

integrals, 0 will receive all the values between 0 and 27r, inclusive of the inferior limit, and exclusive of the superior, and will therefore never actually be equal to, or exceed 2?r. Also in all cases we shall be concerned with the same may be supposed true of <p' ; for in all our investigations, wherever 0' recurs, the value <f)' = 2 IT, or any greater quan- tity, will be only a repetition of <f>' = 0, or some value between 0 and 2 TT ; hence we may consider that <f> ~ <f> never actually equals or exceeds 2-rr; and hence it will be only for one value of 0 ~ 0', viz. 0, that p will become unity ; hence, by what we have proved, the series Q0 + 3Q, + &c.

in all cases we shall be concerned with, will be absolutely zero for all values of fj. and 0, except the single values [A. = p and 0 = 0'. We now proceed to prove some re- markable properties of this series, which result from its discontinuous nature.

26. From Art. 22 it appears immediately that

= 4<7T.

Now, here the quantity under the integral signs is, as we have proved, always zero, except when p. = /m' and 0 = 0'; it is therefore no matter what the limits of the integration be, provided they include between them the

or supposing >rj + ndx = ,r2, df(xt) + df(xl + dx)+&c. till we come to df(xa-dx)=f(xs)-f(xl)

= £«<*/(*).

Hence it appears that if*8 df(x) denotes the sum of a series of values of df(x), got by giving x all its values between the limits xt and xz inclusive of the former limit and exclusive of the latter; that is to say, all the values of x which form an arithmetic series whose common difference is dxy commencing with xl and ending with xz- dx. The remark respecting the limits is im- portant whenever discontinuous functions are concerned, as in our present in- vestigation ; and we must remember that though the last value of x approaches indefinitely near to x2, it never actually becomes equal to it.

alues (fji = /u') and (0 = 0'), respectively ; hence, if /*,, ^2, i» 02*5 be any limits which do this, we have

* (Qn + 3Q! + &c. ...)dnd<b = ±Tr.

27- In the same manner, if F (n<p) be any function of /u and 0, which is always finite between the limits - 1 and 1, 0 and Zir, F(yu</>)(Q0 + 3Q1 + &c.) will be always zero, except when JM = // and 0 = 0', and we shall have, as before,

/^2 rM

•^l »^l

28. Now let F(n"(f>") be the greatest value of F(^0), between the limits /ui, yua» 0i> 02 '•> ano< let ^'(/*//0//) be the least ; then it is evident from the nature of an integral, considered as a sum, that

/-</>* rn* F ^/0/j ^QQ + 3 Q! + &c) d^d0 is not greater than

.F(/u"0") y T 2 (Q0 + 3Q, + ...) dyurf0, and not less than

i. e. (by Art. 22),

not greater than 4 ?r F (^"0"),

and not less than 4 TT T'1 (^^0^) ;

and this is true, no matter how close together the limits

* Of course these limits are supposed to be included between - 1 and + 1, 0 and 2ir.

/"u Us* <£n ^a? be taken, provided /UL' and (j)' be included be- tween them. Now /*", 0", and /u//5 0//s are also always included between these limits ; hence, since /&'<£', M"> </>"» yu//s 0/x, are respectively always included between limits which we may take as close together as we please, it is evident that we may suppose //', 0", and //„» 0y/» to differ from yu'0' respectively by as small quantities as we please ; and therefore, since F (/u. <p) is always finite, we may in the above inequalities suppose F(/JL'> <^>") and F (p.^ <£„) as nearly equal to F(fj!<p') as we please, which evidently cannot be, unless

FT

jfr jfr

29. We shall give another demonstration of this remarkable result.

Assume, as we evidently may, (Q0+3Q1+ ...

Then, as before,

r* T

fc/Q •/ — 1

multiply this equation by d// d^)', and integrate between the same limits, and we have*

It is necessary to take the same limits, otherwise in the integral

/i and ^>, the values of the variables for which Q0+3Qi + ... becomes «, will not be always included between the limits, and therefore Art. 2fi will not apply to it, and our proof will be incorrect.

24 Hence, by Art. 26',

differentiating this equation relatively to <p., and yu2 suc- cessively*, we have

4-TT F (fJL2(f)o) =,/*(Al2</>2);

hence, since ^2 and 02 are arbitrary ,y= 47T-F; and therefore (Q0 + 3Q,+ ...) dfjidQ

30. We may hence find the value of TV' YnQndndd).

Jn J -i

Yn being any Laplace's coefficient ; for Yn being a rational and integral function of /m, \/l-to>2, cos ^>, sin 0-f- will be always finite ; hence we may put Yn for F (/a. (pi) in Art. 28, and we find immediately by Art. 21,

PV YnQn

J0 J-i

* To shew how to differentiate a definite integral with respect to its limits, let f(x) + C denote the indefinite integral off'(x), then

differentiating this relative to <r2, we have

and differentiating relative to

and in a similar manner we may differentiate integrals relative to two or more variables.

t This is the reason why we have restricted Laplace's coefficients to be rational and integral.

25

where Y ' denotes what Y becomes, when /tx.' and <f> are put for (ix and (f> in it.

This is a very important result ; in fact, this, and that in Art. 21, are the properties which render Laplace1 s coefficients so very useful in integrations such as we have to perform in Art. 15.

31. The equation deduced in Art. 28, interchanging fjL and (f)' for /tx and <£, shews that if F(fj.(j)) be any func- tion of fjL and (p, which is always finite, it may be expanded in a series of Laplace's coefficients ; for by this equation F (n<p) = a series whose general term is

2n + 1 rt tr /M , , ,

— / I F (/JL <p ) Qn d fJL U (t) .

4>ir J o »'— i

Now this quantity evidently satisfies any linear differ- ential equation relative to fj. and 0 that Qn satisfies ; there- fore it satisfies Laplace's equation of the wth order ; moreover it is a rational and integral function of /x, \/l — ti2, cos <£, sin<^>, for Qn is so, and

2W+ 1 /-8W /M r

will evidently differ from QB, considered as a function of ^, \/l-/x2, cos 0, sin 0, only in having different coefficients to the powers of these quantities ; that is to say, if A' be any coefficient in Qn, then the corresponding coefficient in

') Q. dp' dip

27T

will be

hence the several terms of the series to which F'(/u'(p') is equivalent are rational and integral functions of /u, \/l-/r, cos 0, and sin 0, which satisfy Laplace's equation, and

4

are therefore,