(Stack >nnex IQB O'BRIEN'S MATHEMATICAL TRACTS, PART I. y^s v&)C;".X .L> I 1. - 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 + w 8 (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 Hymers 1 Geometry of three dimensions, page (77). Then we have off = r sin 9 cos d), y = r sin sin <^>, % = r cos 9, and similar expressions for v' t y', %'. Hence the equation (A) becomes C = V + r- sin 2 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 and r, being the limits of r', and TT of 9', and 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
, 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. r 2 (l /u 2 ) must be always small : therefore since r 2 (l fj?) is not always small, w z 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 to 8 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> 2 w) 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> 2 w'), when u denotes what u becomes when a, /*, and are exchanged for a', /, and
dfi ' w
^ f /(a, a') dp.' d 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+a 2 '
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 Q y , Q n Q 2 , &c. to denote these co-
efficients, i. e. we shall assume
Q will evidently be unity. The rest of these coefficients
will be rational and integral functions of p, i.e. of
fifi + V 1 yU 2 . V 1 // 2 . COS (0 0').
It is evident that they all become unity when p becomes
unity ; for then
becomes , or 1 + h + h 2 + &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,, Q n Q 2 , 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? + (y f - 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 ;
and therefore since x = r sin cos 0, and y = r sin 6 sin 0,
we shall have
X = S COS 0,
y = s sin 0.
Then considering s and as independent variables
instead of x and y, we have
dR _ dR dx dR dy
ds dx ds dy ds
dR dR .
= cos + sm0 (2).
dx dy
d z R d*R d z R d?R
and = cos^0+2 cos sin 0+ sm 2 0...(3).
ds* dx 2 dxdy dy 2
. 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 2 . S^COS Sin H S 2 COS Z (b
d0 dx 2 dxdy dy 2
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 + sin
ds* s 2 d0 2 dx 2 dy 2 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
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 + r 2 drF" ds r~d7"
Now by (2) and (4),
dR dR cos0 dR
sin + 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 r 2 , 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
V r -
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 dO \ d9 ] sin* d(f>
which is a partial differential equation of the second order,
connecting Q n with /a. and ; of course, being such, it
admits of an infinite number of solutions besides Q n .
We shall have no occasion to solve it, but "we shall
find it of use in investigating the properties of Q n . All
solutions of it which are rational and integral functions
of cos 0, sin 0, cos and cos <7>, and not of ~ ' = 0, or some value
between and 2 TT ; hence we may consider that <^>") and F (p.^ <)
as nearly equal to F(fj! , the values of the variables for which Q +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 ir J o ' i
Now this quantity evidently satisfies any linear differ-
ential equation relative to fj. and that Q n satisfies ; there-
fore it satisfies Laplace's equation of the w th order ; moreover
it is a rational and integral function of /x, \/l ti 2 , cos <, sin<^>,
for Q n is so, and
2W+ 1 /-8W /M r
will evidently differ from Q B , considered as a function of
^, \/l-/x 2 , 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 Q n , 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, according to our definition, Laplace's co-
efficients.
32. No function can be expanded in more than one
series of Laplace's coefficients.
For, if possible, let
Y + Y l + + ... + &c. and Z + Z l + Z 2 + &c.
be two different series of Laplace's coefficients equivalent to
the same function, then, since this is the case, we have
Y - Z + Y, - Z l + Y, - Z 2 + &c. = 0;
multiplying this equation by Q n d/i.d are substituted for
o', /n', 0', and therefore we shall have
u = u + u l + u 2 + &c.
We shall also have occasion to make use of the values
of w' , 7/ n w'j>, Sec. when /u. and (f) alone are put for /u.'
and 0', a' remaining unaltered, These values we shall
denote by 'w , f u iy 'w 2 & c -
39.
find
and
J9. Hence in the equation (5'"), see Art. 15, we
immediately by Arts. 22 and 30,
f
32
and hence, putting u + w, + w 2 , &c. for u,
y _ 47r / - y i ^ . 2 _ /.. ... . .. . o \ a ^o
= 4?r / ' p A da
/
+ a series whose general term is
4-7T + y cos h sin
= y sin (h + 0).
Now in the figure (page 66), the angle SZX is A, and
the angle XZC is 0, hence the angle *S*ZC is (f) + h; there-
fore if we take the point A to be the pole of the ecliptic
(which we may do since its position is arbitrary), it is evi-
dent that ^ + h or SZC will be the Sun's right ascension
76
90; hence, if a be the Sun's right ascension, we shall
have
$ + h = a - 90,
and hence our equations become
d^ .
sin 9 = y sin a,
dO
- = - 7 cosa;
or, putting for y its value,
d^ . 3n' 2 Q .
- sin 6 = sin A cos A sin a,
at n
d9 3ri z $ .
= sin A cos A cos a.
dt n
82. Now if / be the Sun's longitude, /, a, and 90 - A
are the sides of a right-angled spherical triangle, the right
angle being opposite Z, and (the obliquity of the ecliptic)
being the angle opposite 90 A ; hence since, by Napier's
rules,
cos A = sin 6 sin I (1),
cos Z = sin A cos a (2),
and sin a = cot A cot 9 (3),
we have
sin A cos A sin a = cos 2 A cot 0, by (3),
= sin cos 9 sin 2 Z, by (l),
and sin A cos A cos a = cos Z cos A, by (2),
= sin 9 sin I cos Z, by (l) ;
hence our equations become