(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 <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. 
 
 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 <p' respectively. 
 
 Assume for brevity, 
 
 then in the equation (B') we shall have 
 
 * = / o ai p/ (r, /) . da', 
 
 where Oj is the parameter of the exterior surface. 
 
 and substituting for r,f(r,r')= t f{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 (B 7 ) becomes 
 
 14. This expression for V is much more manageable 
 than that in the equation (B 1 ) ; 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 ra(l + <o 2 w), which, 
 neglecting the squares and higher powers of <o 2 , 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+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 <f), sin <p (i. e. of /x, \/l-fj? 9 cos (f), 
 sin 0), are called Laplace's coefficients of the n th 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 - M 8 > cos 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 w th order is also a Laplace's co- 
 efficient of the w th order. 
 
 Also any constant quantity is a Laplace's coefficient of 
 the order 0, for if F be a Laplace's coefficient of the order 
 0, we have 
 
 d 
 
 which equation is evidently satisfied by 
 F = 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 - /u 2 cos (0 + /3) 
 
 are Laplace's coefficients of the order 1, a and /3 being 
 constants ; and 
 
 a (1 - M 2 )> a jtA \A ~ M 2 cos (0 + /3), a (l - fj?) cos (2 + /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 w th order, 
 but, to give clear ideas, we shall just state that it may be 
 put in the form 
 
 A M n +4 } (I -fjfyM^costy + aj -I- 4,(1 -ffiM n _ 2 cos(2<p + a 2 ) , 
 &c ............. +A(l- 
 
 When AQ A l SEC. ... a^ a 2 ... &c. are any constants, M n M n _^ 
 &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 Y m and Z n be any two Laplace's 
 coefficients of the m th and w th orders respectively, then 
 
 i I Y m Z n dju. 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 Z M satisfies Laplace's equation, we have 
 
 n (n + 1) *" Y m Z n 
 
 Now integrating by parts, and observing that 1 - M 2 = 0, 
 at each limit we have 
 
 and similarly, 
 
 f"v fz -j r ~ dr.dz 
 J, Y "^ d * = - J. Tp-jf**. 
 
 1 /^ 
 
 * observing that Y m ---* is the same at each limit, be- 
 ct(p 
 
 cause Y m and Z n are functions of sin and cos 0, and not 
 of simply ; hence substituting in (l) 
 
 n.(n + l) r*r 
 
 J J-\ 
 
 r^ r i (/ 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) } * Y m Z n dnd<t>= . 
 
 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 Y m z n d/uid(f>=o, 
 
 J J -} 
 except when m = n. 
 
 22. Since Q = 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 Q , Q 15 Q 2 , &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 
 
 
 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+ 3 Q/+ & 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 + ... Q n h + &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 
 
 
 
 O, + 3 Q + &c. = - = , 
 (2 - 2p)f 
 
 hence QQ+ 3Q 7 + &c. = for all values of />, except p 
 
 
 
 when it becomes - : now 
 
 
 P=fJLfl' + \ -/(Z 2 ^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' is zero, or some multiple of STT; hence 
 the series Q + 3 Q x + &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 
 Q + 3Q l + Sec., it will occur under integral signs relative to 
 fi and 0, and the limits of <f) will be 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(x 3 )-f(x l ): 
 but this notation has a much more important signification : for in the equation 
 
 put for x successively the values .r,, Xi + dn, z 1 + 2dn, &c., #, + (- \)dx, 
 and add the results, and we find 
 
 ! + dx) + d/(-r, + 2dx) &c. + df{ Xl + (n - 1) dx\
 
 21 
 
 integrals, will receive all the values between 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 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', viz. 0, that p will become 
 unity ; hence, by what we have proved, the series 
 Q + 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'. 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'; it is therefore no matter what the limits of the 
 integration be, provided they include between them the 
 
 or supposing >rj + ndx = ,r 2 , 
 df(x t ) + df(x l + dx)+&c. till we come to df(x a -dx)=f(x s )-f(x l ) 
 
 = <*/(*). 
 
 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 x t and x z 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 dx y commencing 
 with x l and ending with x z - 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 x 2 , 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 
 
 * (Q n + 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, and Zir, F(yu</>)(Q + 3Q 1 + &c.) will be always 
 zero, except when JM = // and = 0', and we shall have, 
 as before, 
 
 /^ 2 r M 
 
 ^l ^l 
 
 28. Now let F(n"(f>") be the greatest value of F(^0), 
 between the limits /ui, yua 0i> 02 '> ano< l et ^'(/* // // ) be the 
 least ; then it is evident from the nature of an integral, 
 considered as a sum, that 
 
 /-</>* rn* F ^/0/j ^Q Q + 3 Q! + &c) d^d0 
 is not greater than 
 
 .F(/u"0") y T 2 (Q + 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, 
 and 2ir.
 
 /"u Us* <n ^a? be taken, provided /UL' and (j)' be included be- 
 tween them. Now /*", 0", and /u //5 //s are also always 
 included between these limits ; hence, since /&'<', M"> </>" 
 yu //s /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 // y/ 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, 
 (Q +3Q 1+ ... 
 
 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 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 <p., and yu 2 suc- 
 cessively*, we have 
 
 4-TT F (fJL 2 (f)o) =,/*(Al2</>2); 
 
 hence, since ^2 and 2 are arbitrary ,y= 47T-F; and therefore 
 (Q + 3Q,+ ...) dfjidQ 
 
 30. We may hence find the value of 
 TV' Y n Q n dndd). 
 
 Jn J -i 
 
 Y n being any Laplace's coefficient ; for Y n being a rational 
 and integral function of /m, \/l-to> 2 , cos ^>, sin 0-f- will be 
 always finite ; hence we may put Y n for F (/a. (pi) in Art. 28, 
 and we find immediately by Art. 21, 
 
 PV Y n Q n 
 
 J 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 <r 2 , 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 Laplace 1 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 ) Q n d fJL U (t) . 
 
 4>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<p, and integrating be- 
 tween the limits - 1, 1 ; 0, 27r, we find (by Art. 22 and 30), 
 
 (Y' n -Z' a ) =0, 
 
 "2n + 1 
 
 hence Y n = Z n ; and therefore the series are the same, and 
 the function can be expanded only in one series of Laplace's 
 coefficients. 
 
 33. The conclusion we have just arrived at seems to 
 be at variance with the fact that the quantity 
 
 may be expanded in two distinct series of Laplace's co- 
 efficients, viz. 
 
 but this is only an apparent discrepancy, for 
 
 the function to be developed, admits of two values, one 
 positive and the other negative, on account of the ambiguity 
 of the sign of the square root ; and therefore ought to admit 
 of two developments. That the above are the two develop- 
 ments corresponding to the two values of the square root, 
 will follow from the following proposition : viz.
 
 27 
 34. To determine the sign of the series 
 
 This series being the development of the expression 
 1 
 
 it is evident that it can never become as long as h is not 
 infinite, but it will become infinite when 
 
 1 - 2ph + h 2 = ; 
 
 2 
 that is, when p 
 
 . 
 
 Now 1 + h 2 is always greater than 2A, except when h 1, 
 in which case 1 + A 2 = 2A; hence, since p, being a cosine, 
 can never exceed unity, this equation can only be satisfied 
 by h = 1 , and therefore p = 1 ; hence 
 
 cannot become infinity unless h be unity ; hence, if h be not 
 infinity or unity, this series can never become zero nor in- 
 finity, and therefore can never change its sign : supposing 
 then that h is not infinity or unity, 
 
 will always have the same sign whatever be the value of p. 
 Hence, supposing p = 1, and therefore Q = 1, Q 1 = 1, &c. 
 (Art. 16.) it is evident that 
 
 Qo + Qih + &c. has the same sign as 1 + h + h? ... 
 
 1 
 
 i. e. as - -; 
 1 -h 
 
 hence, if h be less than unity, this series is positive, and if 
 h be greater than unity, it is negative. 
 
 In the same way it may be proved that 
 
 tilc samc 8 ' n as 
 
 \ 7 g -r* 7 - ' 
 
 rtf ft ft -~ 1
 
 and is therefore always negative when h is less than unity, 
 and positive when h is greater than unity. 
 
 Hence, when h is less than unity, 
 
 i 
 QO + Q\h + Qzh 2 + is the positive value of 
 
 V 1 - 2ph + tf 
 
 and - \ Qo + Qi 7 + . . . \ the negative value ; 
 h ( h } 
 
 and when h is greater than unity, the reverse is the case. 
 
 35. To represent, then, the true general development 
 
 of j- , let A; be a discontinuous function of h, 
 
 Vl %ph + h* 
 
 such that A: = 1, when h is less than unity; 
 
 and k = 0, when h is greater than unity. 
 
 Then, whatever h be, the positive value of ^ 
 
 V ] 
 
 will evidently be 
 
 and the negative value will be 
 (1 -k} {Q + Q,h + Q 2 A 2 + ...} + k JQ ^ + Q ^+ ...1. 
 
 Thus, in reality, , , if we restrict our- 
 
 Vl-2ph + h z 
 
 selves to only one of its values, can be developed in only 
 one series of Laplace's coefficients. The conclusion we have 
 just arrived at respecting the true development of this 
 quantity, will be highly important in our future investi- 
 gations. 
 
 36. We shall conclude this chapter with the following 
 important property of Laplace's coefficients, viz.
 
 29 
 
 If Y +Y l + F 2 + &c. = (l), 
 
 be any equation arranged in a series of Laplace's coeffi- 
 cients, IJL and being indeterminate, then we must have 
 
 F =0, Y,= 0, F 2 = 0, &c. 
 
 for, multiplying (l) by Q,,dfj.d(p, and integrating between 
 the limits - 1, 1, and 0, 27r, we have, (by Arts. 22 and 30,) 
 
 2tt + 1 
 and therefore Y n = 0. 
 
 We have now considered these remarkable functions at 
 sufficient length for our present purpose ; we shall recur to 
 this subject in Part u. of these Tracts.
 
 CHAPTER III. 
 
 FIGURE OF THE EARTH. 
 
 37. WE are now prepared to return to the equation 
 (5'"), see Art. 15 ; the properties we have proved La- 
 place's coefficients to possess, will enable us to perform the 
 integrations relative to // and <' with great facility. 
 
 In Art. 16 we stated that we should expand the quan- 
 tity f (a, a'), or / , in either of the series 
 
 Now there is an ambiguity in this quantity, since on account 
 of the square root, it admits of two values, one positive 
 and the other negative ; but, on referring to Art. 6, it is 
 evident that the square root is always supposed to have its 
 positive value; for the distance between $m and $TW', which 
 is expressed by this square root, is evidently taken to be the 
 absolute or numerical distance, without reference to sign ; 
 hence, by what has been proved in Art. 35, neither of the 
 above series will give the true general development of 
 y(, a'), but we must put 
 
 /(a, a') = *. Q + Q + Q 2 + &c. 
 + (1 - fc)a' JQ + Q,^ + Q 2 ^+ &c.}, 
 where k is a discontinuous function of , which is always
 
 unity while a is less than a, and zero when a is greater 
 than a. For brevity we shall take A n to denote the co- 
 efficient of Q n in this series ; i. e. 
 
 and then, we shall have 
 
 38. What we have proved of Laplace's coefficients 
 naturally suggests the advantage of expanding the quan- 
 tity u in a series of Laplace's coefficients ; this may be 
 done, since, on account of the assumption we have made 
 respecting the nearly spherical form of the strata, u can 
 never become infinite ; we shall assume, therefore, 
 
 u = u' + u\ + u' 2 + &c. 
 
 tt'o, M'J, &c. being Laplace's coefficients of the orders 0, 
 1, 2, SEC. and functions of a', /u', and ('. 
 
 We shall denote by u ot u lt u 2 , &c. what w' Q , u ^ 
 u' 2 , &c. ... become when a, yu, and (f> 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* f ' p' { au n + - -. (a A n 'u n )\da . 
 Jo \ n da 2n + 1 do' v J 
 
 40. If we put for A and .4 n their values, viz. 
 
 a' 2 a n+2 a" 
 
 k + (l-k)a', and k + (1 - A?) ^ , 
 
 
 (see Art. 37.) ; the factor of 47rfc> 2 in this general term be- 
 comes 
 
 Now /c is always unity while ' varies between the 
 limits and a, and zero while a varies between the limits 
 a and a l (a being of course never greater than a^ ; hence 
 that part of the quantity under the integral sign which is 
 multiplied by k will not exist except between the limits 
 and a, and that part multiplied by 1 - k will not exist 
 except between the limits a and a l ; hence this integral may 
 evidently be put in the form 
 
 - U JLf* p'a' 2 da'+- r ~ r ( a p (XX n+s ) da 
 
 a Jo ^ (2n + 1 a n+l J r da' y 
 
 t , d f 'u. \ 
 p , \~-. } 
 ^ da \a n - 2 J
 
 33 
 
 We shall, for brevity, denote this expression by 
 
 <*n ", 
 
 <r n being merely a prefix assumed to express a certain 
 operation performed on W B . 
 
 In the same way the first term of F, viz. 
 
 4-Tr Ip A da', 
 
 "0 
 
 will evidently become 
 
 4>7r Ty { k + (l - k) a] da, 
 
 or 
 
 4>Tr J p a' 2 da + 4nr f l p a da . 
 Hence the development of V becomes finally 
 
 F = f a pa 2 da' + 4>7r / ' pa da 
 
 a Jo r J a ( 
 
 4- 4-TTca 2 cr w + ff\^\ + cr 2 w 2 + Sue. 
 
 It is evident that this is a series of Laplace's coefficients, 
 the sum of the first three terms being a coefficient of the 
 order 0, and the succeeding terms of the order 1, 2, 3, Sec. 
 respectively. 
 
 41. We shall now express the value of Fgot from the 
 equation (A") (see Art. 13) in Laplace's coefficients; to do 
 this we observe, by trial, that -^ /m 2 is a Laplace's co- 
 efficient of the order 2 ; hence the value of V got from (A") 
 will be arranged in Laplace's coefficients by simply putting 
 it in the form
 
 34 
 
 42. If we now subtract the two equations we have 
 thus obtained, we shall eliminate V and arrive at an equa- 
 tion consisting of a series of Laplace's coefficients, which, 
 by (Art. 36), must be put separately equal to zero; hence 
 we get the following equations, viz. 
 
 'o'^d' ' 
 
 l pada' + 4>Tra) 2 (r u C + - = 0, 
 
 a Jo ' J a ' 2 
 
 (^ - fjf) = 0, 
 
 and a- n u n = 0, for all values of n except and 2. 
 
 Thus we have eliminated V without knowing what 
 function it is, and obtained equations for determining w , 
 MJ, u Z9 &c., and therefore the equation to any stratum. 
 
 43. We shall now proceed to solve these equations, 
 commencing with the last of them, viz. 
 
 <r n u n = O...(l), except n=0 or 2. 
 By (Art. 40) this equation is equivalent to 
 
 -^ r a p f a rt da' + - - l - - f a p-^.(a' n +* f u n )da 
 a Jo ^ (2n + l)o" +l ./o ^ da 
 
 a" ro, , d ( '. \ , 
 
 / P -r-7 -r-i ) da = - 
 1 Jo r da k*~y 
 
 + 
 2n + 
 
 Now, by Note, p. 24, if we multiply this equation by a w+1 , 
 and differentiate it relatively to a, and then multiply it 
 
 by , and differentiate it again relatively to a, we shall 
 
 by this process get rid of the two last integral signs, and 
 arrive at a differential equation which may be put in the 
 form 
 
 du n 
 A +4...0 ...... ( 2 ),
 
 35 
 
 A lt Ay being functions of a and p, which we have no oc- 
 casion to determine. Now let v and v' be two quantities 
 which satisfy the equations 
 
 cr n v a n = 
 
 1 ......... (3), 
 
 and let C and C" be two constants, then 
 
 u n = Cv + CV 
 
 will be a solution of the differential equation (2) ; for, per- 
 forming the operation a n on both sides of the equation, 
 
 u n = Cv + CV, 
 we have & n u n = cr n (Cv + CV) 
 
 = C(T n V + C'(T n v' 
 
 (evidently from the nature of the operation <r n ) 
 or ff n u n = Ca" + C' ^ , by (3). 
 
 Now if we get rid of the integral signs in this equation by 
 the same process we have applied to the equation (1), the 
 second side will evidently be made zero by the differenti- 
 ations, and thus we shall arrive at the same differential 
 equation as before; hence 
 
 ,-Ca + CV 
 
 is a solution of (2). Now this is evidently true whatever 
 be the values of C and C"; hence this solution contains 
 two arbitrary constants, and is therefore the most general 
 solution that (2) admits of. Hence all values of u n which 
 satisfy (l), since they also satisfy (2), must be values of 
 
 * It is evident that v and v' are two different functions, otherwise the 
 equations (3) would give 
 
 which is not the case. This remark is necessary, for if v and t>' were not 
 different functions then C and C' would add together, and therefore be equi- 
 valent to only one constant.
 
 Cv + C'v. To determine what values of Cv+C'v' satisfy 
 (1), substitute Cv + C'v for u n in (1), and we find 
 
 or Ca" + C' = 0, by (3). 
 
 Now this equation ought to be true for all values of a ; 
 hence (7 = 0, and C'=0; hence it is evident that only 
 one value of Cv + C'v, namely zero, satisfies (l); and 
 hence it follows from the equation (l), that 
 
 Thus u n is zero for all values of n except and 2 ; this 
 result produces a considerable simplification in the equation 
 to the strata. 
 
 44. We shall next consider the equation involving w 2 , 
 which may be written thus, 
 
 In this equation we may conceive w 2 , which is a function 
 of n, vi - ^, cos 0, and sin 0, to be developed in a series 
 of powers of i /j? and <. Let y (^ yu c ) m 0" be any term 
 in this development, -y being an unknown coefficient to be 
 determined, then the corresponding term in the equation 
 will be 
 
 except m = 1 and n = 0, in which case it will be 
 
 Now i /a? and are arbitrary ; hence we must put the 
 coefficients of their several powers separately equal to zero ; 
 and hence for all terms, except that in which m = 1 and 
 n = 0, we have 
 
 0"27 = i and therefore y = 0, as before in (Art. 43) ;
 
 37 
 
 and for the term in which m = 1 and n = 0, 
 
 a 2 
 ff *V + ^. = \ 
 
 which equation will determine y ; hence we have 
 
 w 2 = 7 (i ~ M 8 ) 
 where -y is given by the equation 
 
 45. We cannot determine U Q from the remaining equa- 
 tion in Art. 42, on account of the. unknown constant C in- 
 volved in it ; but the value of u follows from the assump- 
 tion that we have made respecting a in (Art. 12) ; namely, 
 that it equals the radius of the sphere of the same capacity 
 as the stratum whose parameter it is: for this assumption 
 gives 
 
 => volume included by stratum, 
 3 
 
 rtir r 1 / 
 
 I. Li 
 
 a 3 rz-T 
 
 jjf 
 
 a(l+a> 8 u) 
 
 47ra 3 a> 8 w > by (Art. 22.) 
 
 Hence U Q = 0. 
 
 46. Thus we have determined the values of w , M,, 
 w 2 , &c. ; and it now only remains to substitute these values 
 in the equation, 
 
 r = a . { I -i- or (M O + u t + u 2 &c.) | ;
 
 38 
 
 and we find that the equation to any stratum whose para- 
 meter is a, is 
 
 Now the equation to a spheroid generated by an ellipse 
 revolving about its minor axis, is 
 
 V 1 - e 2 cos a l^ - 9\ 
 
 a being the major axis, e the eccentricity, and ,the angle 
 which r makes with the minor axis. Supposing e very 
 small, this equation becomes 
 
 e 2 e~ 
 r = a \l + ~sin 2 0* 
 
 2 2 
 
 = a {l -cos 2 0j, 
 e being the ellipticity, 
 
 maior axis minor axis 
 
 for the ellipticity = ? : : 
 
 major axis 
 
 -, nearly. 
 
 This equation may evidently be made to coincide with 
 the equation to the stratum, by putting 
 
 a e = o to 2 ; 
 
 to 2 '/ 
 and therefore e = - :: = ft) 2/ Y nearly.
 
 Hence the strata are all spheroids of revolution about 
 the polar axis ; the ellipticity of any stratum being ufy, 
 y being got from the equation 
 o 2 _ 
 
 STT 
 
 or putting in this equation for y, the equation for deter- 
 
 (t) 
 
 mining the ellipticity of any stratum, will be 
 
 8?r 
 or, by Art. 40, 
 
 / p a da' H / p' "' (a' 5 e')da r 
 
 a Jo 5a Jo da 
 
 a 8 /-"i ,de 0*0? 
 
 + p ; da + = 0, 
 
 5 Ja ^ da' STT 
 
 which equation will give e when p' is known as a function 
 of a'. 
 
 47. Thus we have arrived at the remarkable result 
 that the mass must be arranged in strata of equal density, 
 which are all spheroids of revolution about the axis of rota- 
 tion, their ellipticities being connected by the equation just 
 obtained. 
 
 It is evident that our investigation gives us no infor- 
 mation respecting p ; hence the law of density of the strata 
 is quite arbitrary, and must be determined, if possible, by 
 some independent method. 
 
 48. We shall presently shew that the results of our 
 investigation may be compared with observation in a most 
 satisfactory manner, without knowing any thing of the law 
 of density of the strata. This is fortunate, as we have no 
 means of determining this law, but must have recourse to an 
 hypothesis which it must be confessed is rather empirical ; 
 but as the results it leads to may be made to agree well with
 
 observation, we must look on it as probable. The hypo- 
 thesis we allude to is this, that the variations of the pressure 
 in the interior of the earth (supposed fluid and of the same 
 chemical constitution all through,) are proportional, not to 
 the variations of the density as in gases of uniform tempe- 
 rature, but to the variations of the square of the density ; 
 i.e. that instead of having dp = kdp, we have dp = kpdp. 
 There is some slight reason a priori for assuming this 
 formula, for it is evident that p ought to increase more 
 rapidly with p in the fluid composing the earth, than it 
 would do in gases, both on account of the incompressibility 
 of that fluid, and the increase of temperature as we go 
 towards the centre ; and hence kp dp will represent the 
 variations of p better than kdp. But the chief reason for 
 assuming this formula is, that it leads to correct results, and 
 simplifies the equations we shall be concerned with, as will 
 appear. 
 
 49- Assuming then this connection between the pres- 
 sure and density, we may calculate the law of density from 
 the equations 
 
 .................................... (1), 
 
 V = ^-J^ p'a'da'+tTT r i p'a'da ......... (2); 
 
 which are got from (Art. 6) and (Art. 42) ; neglecting or, 
 for the centrifugal force will make a very little difference 
 in the law of density, and it will be useless to be very 
 accurate here, as we are proceeding on rather uncertain 
 grounds. 
 
 We have, from (l), 
 
 dp_ dV 
 d^** P da
 
 multiplying this by , and differentiating relatively to a, 
 
 P 
 we have 
 
 d 
 
 Substituting the assumed value of dp, viz. dp kpdp, 
 we have 
 
 which may be put in the form 
 
 Therefore, putting = g 8 , 
 
 pa = A sin (70 + 5), 
 sin 
 
 and p = 
 
 A and B being arbitrary constants. 
 
 50. To determine B, let a = 0, then we have 
 p = co , unless J9 = ; 
 
 hence since p, as we may evidently assume, is not infinite 
 at the centre, B = 0, and we have 
 
 A sin qa 
 P ~ a 
 
 To determine A and q, let pj be the density of the superficial 
 parts of the earth, i. e. the value of p when a a, , 
 and let D be the mean density of the earth ; then 
 
 mass of earth 
 
 its volume 
 6
 
 47T 
 
 42 
 
 / pa* da' 
 
 JQ 
 
 s 
 
 3 A 
 
 3A ra, . 
 
 = I a sin q a da ; 
 a 3 J 
 
 "T ^ sin go' 
 
 3A 
 
 - . 
 
 also 0i = sin 
 a, 
 
 hence 
 
 sin 9^! qa l cos graj 9 a\ JJ 
 
 from which equation q may be found, and then we shall 
 have A from the equation 
 
 sin 9 a, 
 
 51. Observation shews that is about , and on 
 
 D 11 
 
 substituting this value of --j- in the equation for deter- 
 mining (7, we shall find by repeated trials (which is the 
 only way we can solve it) that it admits of several solu- 
 tions, of which one only leads to right results; it is this, 
 
 <7i = about ; 
 
 1 5-7T . 
 
 hence, substituting for (7, we have 
 
 . . /57T a 
 A sin | 
 o a.
 
 43 
 
 To determine A, we shall put a = a, in this, which gives 
 
 A . 5-n- 
 
 sin T 
 
 - sin 30 
 
 and hence .4 = S^e^, and we have 
 
 O 
 
 which, if our hypotheses be correct, expresses the density 
 of any stratum in terms of the parameter of that stratum, 
 and the superficial density. 
 
 52. The method by which we have arrived at this 
 formula for the density is not very satisfactory, and we 
 shall therefore consider it as empirical ; we observe that it 
 gives a density which increases as we go towards the centre, 
 but does not become infinite there; this is most probably 
 the case; it also makes the pressure vary more rapidly as 
 we approach the centre than it would do if the earth were 
 gaseous and of uniform temperature; this is also most 
 probably the case; and it gives the mean density of the 
 earth its proper value: we shall prove presently that it 
 gives the true value of the earth's ellipticity, and also the 
 true value of the coefficient of precession; hence, on the 
 whole, we may assume it with some probability as the law 
 of density. 
 
 53. Hence, finally, it follows from the hypothesis of 
 the earths original fluidity ; 
 
 (1) That the earth ought to consist of equidense 
 strata, all spheroids of revolution about the axis of 
 rotation.
 
 44 
 
 (2) That if e be the ellipticity of any of these strata 
 it satisfies the equation 
 
 - i/vw + A/V/-- < vv' 
 
 a Jo r 5a?Jo r da 
 
 a 2 /">i ,de . , a 2 
 ' 
 
 a /">i ,e . , a <t) 
 + / p , da'+ - = 0. 
 5 > r da STT 
 
 (3) That we may assume with probability the law of 
 density to be 
 
 a /STT a\ 
 
 p = 2/0, - sin - ) . 
 ri a t \ 6 aj
 
 CHAPTER IV. 
 
 METHODS OF COMPARING THE RESULTS JUST ARRIVED 
 AT WITH OBSERVATION. 
 
 54. IN order to test the correctness of the conclusions 
 we have just arrived at, we shall now deduce from them 
 results of a more practical character, which shall admit of 
 direct comparison with observation : the first result we shall 
 deduce is this ; 
 
 If * be the length of a meridian arc, measured from the 
 pole to any place whose colatitude is c, then 
 
 a and e y being the major axis and ellipticity of the earth. 
 For, by Art. 48, the equation to any meridian is 
 r = a (1 - e, cos 2 0) 
 
 = (l--cos20) ......... (1). 
 
 XT ^^ 
 
 Now_= r* 
 
 = r, neglecting squares &c. of e, 
 
 _{!-!.<_ | COg 20} by (,), 
 
 therefore, integrating, 
 
 (2).
 
 46 
 
 Adding no constant because evidently = when 
 = 0. 
 
 Now c (the colatitude) is the angle made by the normal 
 with the polar axis ; hence (9 c) is the angle made by the 
 normal and radius vector, and hence 
 
 or, neglecting the squares &c. of small quantities, 
 9-c = e sin 20, by (1), 
 
 therefore 
 
 9 = c + e, sin 2 9 
 = c + e t sin 2 (c + e sin 20) 
 = c + e / sin 2c nearly ; 
 
 hence, substituting in (2), we have 
 
 a \c + e sin 2c -- - c - sin 2c|, 
 2 4 
 
 55. To shew how this result may be compared with 
 observation, let s' and c' be the values of s and c correspond- 
 ing to another place near the former, and on the same 
 meridian, then 
 
 and therefore 
 *' - s = a 
 
 j 1 1 - |) (c' - c) + ^ (sin 2c' - sin 2c)| , 
 
 cos ''+ c> sin c '- 6 ''
 
 47 
 
 Now in this equation s' s, being the distance between 
 two places near each other, may be determined by the usual 
 method of triangulation ; cos (c + c) may be found by any 
 of the ordinary methods of determining the latitude of 
 places, without aiming at any great accuracy, since it is 
 multiplied by the small quantity e ; c' c, not being mul- 
 tiplied by a small quantity, must be determined more ac- 
 curately by observing the meridian zenith distances of the 
 same star at the two places, and taking the difference 
 which will evidently be equal to c c ; thus we may put 
 our equation in the form 
 
 when A, B, C are known ' quantities got by observation. 
 
 In the same way, by observations at other places, we 
 may obtain any number of similar equations; suppose 
 them to be 
 
 &c ....... &c ....... 
 
 From any two of these equations we may determine 
 a and e, ; and the values of a and e so determined* ought to 
 satisfy all the other equations; hence, if we find that all 
 these equations are satisfied by the same values of a and e y , 
 it is evident that our result agrees with observation. 
 
 56. Now a number of meridian arcs have been mea- 
 sured, and a system of equations similar to the above have 
 been formed, and it is found that the values 
 
 a = 3.Q62.82 miles, e = , 
 
 ' 306 
 
 satisfy them all to a remarkable degree of accuracy, allowing 
 for certain small errors which may be easily accounted for ; 
 and which, even considering them in the most unfavourable 
 point of view, are very much smaller than e,, which is itself 
 
 * Or rather, the values of a and e, got from all the equations by the method 
 
 of least squares.
 
 48 
 
 a very small quantity ; and indeed if we bear in mind the 
 delicacy and number of the observations requisite in order 
 to form the above equations, the smallness of the errors 
 is most remarkable. 
 
 57. On the whole we are justified in concluding from 
 observation, that the equation to any meridian and there- 
 fore to the earth's surface is very nearly this, viz. 
 
 r = a { 1 - e/ cos 8 0}, 
 where a = 3902.82 miles, 
 
 and . 
 ' 306 
 
 Hence the hypothesis of the earth's original fluidity 
 leads to a very peculiar result, which is capable of varied 
 and extensive comparison with observation, and which 
 agrees with it in a remarkable manner; from which we 
 must conclude that this hypothesis is most probably true. 
 
 58. The second result we shall deduce from our 
 theory is this ; 
 
 If g be the force of gravity at any place whose colati- 
 tude is c, then 
 
 g = U [ 1 + I 61 I COS'C^ . 
 
 Where G is a constant, namely, the value of g at the 
 equator, and m the ratio of the centrifugal force to gravity 
 at the equator*. 
 
 * To determine w, we observe that 
 
 G - 32.2 + a small quantity, 
 hence 
 
 ura 
 
 32.2 + a small quantity 
 
 In putting 32.2 for G we have assumed a foot to be the unit of length, 
 and a second the unit of time; hence we must express a and to 2 in terms of 
 these units, and therefore we have 
 
 a = 3962 miles, nearly, 
 
 = 3962 x 5280 feet, 
 
 and
 
 49 
 
 To prove this, we observe that -g is the resultant 
 of the forces which act on a particle at the surface, which 
 forces are, by (6), 
 
 dV dV dV 
 
 + ftT<r, - + ory, , 
 das dy dz 
 
 and therefore g must balance these forces ; hence, by the 
 principle of virtual velocities, if dr be any variation of 
 r, and da?, dy, d#, its resolved parts along the axes of 
 co-ordinates, and >// the angle which r makes with the 
 direction of g, we shall have 
 
 .r ) dx + ( -- \- u?y }dy + dsr, 
 \dy / * dx 
 
 which gives 
 
 I 
 \d.T. 
 
 " 
 
 r 
 
 Now g acts in the normal (by the principles of 
 Hydrostatics), and the normal evidently makes a very 
 small angle with the radius vector ; hence \|/- is very small, 
 
 V 
 
 and therefore, since cos \J/ = 1 - + &c., we may, neglect- 
 
 ing squares and higher powers of small quantities, put 
 cos \js = 1 ; moreover we have, by (6) and (42), 
 
 and 
 
 4-Tr r<i , , n , , /*", , . , , w 2 a 2 
 
 C = / pa* da + 4-TT p a da + - ; 
 a Jn f Ja ' 3 
 
 Htld " > = 24ho^ 
 2-r 
 
 24x60x60' 
 
 Mnkins; the<:e substitution*, we shall find 
 m = about *
 
 50 
 
 hence, observing that a is a function of r in virtue of the 
 equation 
 
 r = {l+e(i-M 2 )}, 
 we have, substituting in (1), 
 
 dC 
 
 g= -dr 
 _dCda 
 dr dr 
 *TT ra , , 2ft, 2 a\J d(ae) \ 
 
 a da l ~ - ( - M 
 
 which, neglecting the product of small quantities, may 
 evidently be put in the form 
 
 where 
 
 G ** - I p'a' a da' + small quantities. 
 
 Since g *= G when /u = 0, it is evident that G is the 
 force of gravity at the equator. 
 
 (Of course, in all our formulae, a t is supposed to be 
 put for a after all differentiations and integrations have 
 been performed, since the particle on which g is the force 
 is supposed to be at the surface.) 
 
 Now by (53), 
 
 -1 ( a p ' a ' z da'+~ ^ o'-^ 
 a Jo P 5a 5 Jo r da 
 
 a 2 /*i ,de , , 
 + / P -r-, ; da + - = 0. 
 5 Ja ? da STT 
 
 Multiplying this equation by a 3 , and differentiating relative- 
 ly to a, we have 
 
 d(ea?) ra , , r a, ,de , 5ft, 2 a 4 
 
 - / oVW+a 4 / o ,da'+- - = 0, 
 da Jo r J" r da STT
 
 51 
 
 or, observing that a f is to be put for a, 
 
 8?r / p a> da 
 _ 5g,V 
 
 = TcT' 
 
 since G = ~ f" pa'*da + small quantities, 
 
 fl ''O 
 
 5ma 
 
 = 5 
 
 centrifugal force a> 2 o 
 
 since m = ; = - ; 
 
 gravity at equator G 
 
 hence = - e 
 
 da a da 
 
 5m 
 and hence equation (l) becomes 
 
 since = c + esin2c, by (64). 
 
 59- To shew how to test this result by observation, 
 we observe that if p be the length of the seconds pendulum, 
 
 then since 1 = 27T \/- , we have p = ^L 
 
 x-i 
 
 when P = 2 = value of j at the equator.
 
 52 
 
 Now p may be determined by observation at any place ; 
 hence, by observations at different places, we may find a 
 system of equations such as before ; viz. 
 
 See. &c. 
 
 when A y B, A', B"> SEC.... are quantities got from observation. 
 Now it is found that the values 
 
 , 5m 
 P = 39.01228 inches, and -- e/ = .005321, 
 
 satisfy all these equations, not so exactly as before, but yet 
 with remarkable accuracy, considering the small quantities 
 we are engaged with; hence this result is another proof 
 of the Earth's original fluidity. 
 
 60. The comparison of this result with the former is 
 a strong additional proof; for the former result gives 
 
 and since we know that m = - , the present result gives 
 
 this is a very remarkable coincidence, and must be consider- 
 ed as a decided proof of the correctness of our hypothesis. 
 
 61 . If G' be the value of g at the pole, 
 G'=G
 
 _ 
 and 
 
 53 
 G'- G 5m 
 
 / 
 
 This result is Clairaut's theorem. 
 
 62. We may determine the Earth's ellipticity by means 
 of the law of density assumed in Chap. in. 
 
 The equation for determining e is 
 
 - 1 f'p'aPda' + ~ f V J~, (ea*)da f 
 aJ r 5a s J r da 
 
 a /, ,de' , r a* 
 + -/ P^-7da'+ -- = 0. 
 5 Ju r da 8-rr 
 
 Integrating the second and third terms by parts, we 
 have 
 
 a 2 r^du , , , 
 
 
 a r^u , , , of a 
 -- / , e da + L - - 
 5 Ja da 5 
 
 5 Ja da 5 8?r 
 
 A sin qa 
 Now putting p = , we have 
 
 / p'a'~da'= (sin qa qa cos qa) 
 
 dp A , 
 
 -1-7 = -- - (sin q a - qa cos qa ) ; 
 
 hence, if we assume 
 
 e (sin qa - qa cos qa) = ;, 
 (1) will become 
 
 At) A ra , t , A a* /-a, 1J. . , , O. (O*\ , 
 
 --5-+ ,/ via*dc!+ - -da+(^+ a 2 = 0; 
 (f- a 5a i Jo 5 Ja a'* \ 5 S-rrJ 
 
 dividing this by a 2 , and differentiating relatively to a, we 
 have 
 
 A d / ri\ A r a , , 
 - - [-J - - / n a da '= ; 
 tfda\\arj erJ*
 
 54 
 or multiplying by a 6 , and differentiating again, 
 
 63. To solve this differential equation we shall assume 
 ' such a function of a that 
 
 therefore 
 
 Hence, substituting in the equation (2), we have 
 a??-3a fVr'da' + (fa /V Fa'? da'* ; 
 
 /0 ''O 'O 
 
 or dividing by a and differentiating, 
 
 *g-i;+ *".f *?': 
 
 or again dividing by a and differentiating, 
 
 hence <= C sin (qa + C'), C and C' being arbitrary 
 constants, and 
 
 r a'^da = {sin (qa + C') -qacos(qa + C')}, 
 and 
 
 {"a f a a<gda' = {3 [sin (qa + C') - qa cos (qa + C')]
 
 55 
 
 Hence we shall have, putting C instead of , 
 rj = C | -y-j [sin (qa + C')-qa cos (qa + C")] - sin (qa + C')> . 
 
 64. We might determine the constants C and C' by 
 substituting this value of rj in (l), Art. 62, but the following 
 method will be more simple : in the first place we may see, 
 a priori, that C' must = 0, for otherwise we should have 
 /, and therefore e very large when a is very small, contrary 
 to our assumption of the nearly spherical form of the strata ; 
 hence 
 
 rj CJ-jj (sinqa - qacosqa)- sin qa > ...... (2), 
 
 and therefore, since rj = e (sin qa qacosqaj, 
 
 . -- 
 
 q 2 a 2 sin qa - qa cos qa) 
 
 i s 2\ 
 
 We shall determine C by means of the value of ^ ea ' , 
 
 da 
 
 got in p. 51. 
 
 Multiplying (2) by a 2 , and differentiating relatively to 
 a, we have 
 
 d(r,a 2 ) 
 
 = Ca (sin qa qa cos qa), 
 
 da 
 
 also doing the same to the equation 
 
 tj = e (sin qa 70, cos qa), 
 we have 
 
 ; = i (sin qa - qa cos qa) -f eg 2 a 3 sin qa. 
 
 da da 
 
 Equating these values of , we have 
 
 da 
 
 1 d (ea 8 ) sin qa 
 
 '+^0*^ 
 
 oa sin ga - qa cos 70
 
 56 
 
 . d(ea 3 ) 5 ma 
 Hence putting a = a , and --- - = - * , see p. 51, 
 
 5m 3 6/ p, 
 
 = T H -~D~' 
 
 also putting a = n f in (3), we have by Art. 50, 
 
 from which two equations we get 
 5m 
 
 "i" 
 
 H- D 
 
 XT 1 5ir p 5 
 
 rsow m = , qa = , = , 
 
 289 ' 67) 11 
 
 substituting these values and reducing, we find 
 
 e = about . 
 306 
 
 This result agrees with observation, but the agreement 
 is not of much value on account of the assumption of the law 
 of density. 
 
 65. The effect which the attraction of the Earth has 
 on the Moon's motion, is usually brought forward as an- 
 other means whereby we may test the correctness of the 
 hypothesis of original fluidity ; and the agreement between 
 theory and observation in this particular is considered to 
 afford additional evidence of the truth of the hypothesis. 
 We shall attempt however to prove that this is not the case. 
 To do this, we shall shew that if the equation to the Earth's 
 surface be known, and also the law of variation of the force 
 of gravity, then the effect of the Earth's attraction on the 
 Moon follows as a necessary consequence, independently 
 of any theory except that of universal gravitation.
 
 57 
 
 66. It is evident from the smallness of the variations 
 of the force of gravity, that the Earth must consist of 
 nearly spherical strata; hence all the results we have al- 
 ready obtained, so far as they depend on the nearly spheri- 
 cal form of the strata, will be true whether the hypothesis 
 of original fluidity be correct or not. 
 
 Hence, as before in p. 49, we shall have 
 
 putting a / for r in the small term. 
 
 * Now the expression for V in Art. 40. may evidently be 
 put in the form 
 
 a series whose general term is 
 
 + 47TO) 2 
 
 , d 
 
 hence, differentiating relatively to r, and observing that a is 
 a function of r in virtue of the equation r = a (l + o> 2 w), and 
 supposing that a t is put for a after the differentiations, we 
 have 
 
 flTF__47r /. Ol , , /47r \ da 
 
 ~a7~~^J pa 
 
 * For the part of V which is multiplied by 4ir p'a' e da' is evidently 
 ,&c.)}, which =-.
 
 58 
 a series whose general term is 
 
 ( a series wnose general term is \ 
 
 1 (da 
 
 <. n + 1 ra, d , , , , > , 
 
 \--r - -= / P ,(u n a*+ 3 )da + pauA dr 
 I 2n+l n+2 Jo r da' v ' ' ' ") 
 
 , 
 
 + 47ro> z 
 
 - 
 
 (2n+l) < 
 
 Now since r = a (l + ct> 2 w), 
 
 c?a 
 and therefore =1 - &> 
 
 da 
 we have, evidently, neglecting o> 4 &c., 
 
 MTT \ da _ f 
 
 \r t ?' ' * ') dr P ' ' 
 
 and therefore, neglecting a> 4 &c., 
 
 4--?r> w 
 
 a series whose general terra is 
 
 47TO) 2 
 
 hence if, for brevity, we put 
 
 shall have 
 
 -o> 2 ai (1 -/x 2 ); 
 and hence 
 
 47Tft) 2 , 2 3 
 
 5- { z a + - z, + -5 z 2 + & C . 5
 
 59 
 
 Now if we suppose rj and g known, it is evident that 
 the first member of this equation will be known, and may 
 therefore be supposed to be expanded in a series of known 
 Laplace's coefficients ; and hence, since the Laplace's co- 
 efficients of different orders on each side of the equation 
 must be separately equal, by Art. 36, the values of Z Z l Z 2 
 &c. will be known. 
 
 * Now it is evident immediately, from Arts. 14. and 40, 
 that the value of V for any external point is 
 
 V= f ai p'a'*da + 47rte) 2 (- Z + - - Z, + 4 Z 2 + &c 
 r Jo [T r r 
 
 hence, since Z Z t Z 2 SEC. are known, as we have just 
 proved, it is evident that the value of V for any external 
 point is also known. 
 
 67. Hence, if we know the form of the Earth's sur- 
 face, and the law of the variation of gravity, we shall 
 know the value of V for any external point, and there- 
 fore be able to determine the attractions of the Earth on 
 that point without making use of the hypothesis of original 
 fluidity. 
 
 Hence it follows that if the form of the Earth's surface 
 and the law of variation of the force of gravity, calculated 
 on the hypothesis of original fluidity, agree with observa- 
 tion, then the effect of the Earth's attraction on the motion 
 of any external body, such as the Moon, calculated on the 
 same hypothesis, must also agree with observation, whether 
 that hypothesis be true or not; and hence we conclude, 
 that the motion of the Moon does not afford additional 
 evidence of the Earth's original fluidity. 
 
 * For the only difference made in the reasoning in Art. 40, by using the 
 expression for V given in Art. 14 instead of that given in Art. 15, will be 
 
 simply this, that we shall have to consider instead of ; also, since the attract- 
 ed point is external, and therefore r always greater than a', it is evident that k 
 will be always unity.
 
 In the next chapter we shall determine the equations 
 of motion of a rigid body round its centre of gravity, and 
 thence deduce the Earth's motion round its centre of gravity. 
 We shall find that the result affords a confirmation of the 
 law of density assumed in Chapter in., and also of the 
 hypothesis of original fluidity.
 
 CHAPTER V. 
 
 EQUATIONS OF MOTION OF A RIGID BODY ROUND ITS 
 CENTRE OF GRAVITY. 
 
 68. WE know from the principles of Dynamics, that 
 a rigid body acted on by any forces moves relatively to 
 its centre of gravity, in the same manner as if that point 
 became fixed, all other dynamical circumstances remaining 
 unaltered; hence, whenever we wish to investigate the 
 motion of a body relatively to its centre of gravity, we 
 may consider that point as fixed, and this will render the 
 investigation simpler. 
 
 Suppose, then, that we have a body whose centre of 
 gravity is fixed, acted on by any forces ; let m be any ele- 
 ment of it, any % the co-ordinates of Sm at the time t referred 
 to any arbitrary rectangular axes fixed in space, and origi- 
 nating in the centre of gravity ; let L M N be the moments 
 of the impressed forces round the axes of a? y % respectively, 
 then we have, by the principles of Dynamics, 
 
 tfco 
 
 - * - - U M 
 
 (A). 
 
 69. In order to perform the integrations denoted by 
 2 in these equations, we shall introduce new variables 
 instead of xyz, which shall have reference merely to the-
 
 position and motion of the whole body, and not to any 
 particular particle of it. To do this, 
 
 Let x y' % be the co-ordinates of $m referred to any 
 arbitrary rectangular axes fixed in the body, then we have 
 x = se cos (x x) + y cos (y' x) + % cos (% x) ; 
 
 differentiating this relatively to #, and observing that x y %' 
 do not vary with #, we have 
 
 dot , d cos (a?' a?) ,d cos (y' x) , d cos (z'x) 
 
 dt = * dT-~ + y ~ dT~ +% ~~dt ' 
 
 Now the axes of x' y % are perfectly arbitrary ; we 
 may therefore suppose them so chosen that they shall coin- 
 cide with the axes of x y % at any instant we please. 
 Suppose therefore that this coincidence takes place at the 
 time #, then we shall have x' = x, y = y, % = % ; and if,. 
 
 d cos (x'x) d cos (y'x) 
 
 for brevity, we denote the values of -^ - , - y - ,. 
 
 at at 
 
 ~ -, at the instant of coincidence by X X' X" re- 
 
 dt 
 
 spectively, our equation becomes 
 
 dx 
 
 = \x + \y + \ #; 
 at 
 
 X X' X" are evidently variables which have reference to the 
 position and motion of the whole body, and not to any 
 particular particle of it, for they depend simply on the 
 angles which the two systems of co-ordinate axes make 
 with each other at any time, or rather upon the rate at 
 which these angles are varying at the instant of coincidence 
 of the axes. 
 
 In the same way we may prove that 
 
 dy 
 
 -j~ t = M + ft % + fji 'x, 
 
 dss 
 
 - = vss + i >x 4 v y; 
 at 
 
 where /M // n", v v v" are quantities similar to X X' X".
 
 63 
 
 Now since $m is rigidly connected with the origin, we 
 have 
 
 x z + if + s? = constant, 
 
 dx dy dx 
 
 and therefore x +y + ss =0; 
 dt dt dt 
 
 dx dy dz 
 
 substituting in this equation the values of -~- 
 
 dt dt dt 
 
 just found, we have 
 
 Xa? 2 + fiy 2 + v%* + (X'-f fj.")xy+ (p +v")yz + (v + X'')#a? = 0; 
 
 hence, since xyz are arbitrary, we have 
 
 X' + JM" = // + v" = v ' + \" = 0. 
 
 . dx dy dss , 
 
 Hence the values of -p- become 
 dt dt dt 
 
 - 
 dy 
 
 dx 
 
 =v y-Xx. 
 
 To conform to a common and convenient notation, we 
 shall put o>j ft> 2 0)3 instead v" X" /^L" respectively, and write 
 these equations thus, 
 
 dx 
 
 .,-* 
 
 - = W3 ar- tt ,2f 
 at 
 
 we shall presently determine what ojj w 2 o> 3 are. 
 
 These equations express the relations which exist be- 
 tween the velocities of any element of the body and its co-
 
 64 
 
 ordinates at the time , in consequence of the rigidity of the 
 
 body. The substitution of these values of 
 
 at at at 
 
 in the equations (A) will be very advantageous, since 
 o>i c 2 ft> 3 are independent of x y z, and may therefore be 
 brought outside the integral sign 2. To perform this 
 substitution we have from the equations (J5), differentiating 
 
 dx dy dz . 
 
 and m the result putting for their values given 
 dt dt dt 
 
 by the equations (J5), 
 
 d<o 3 
 
 Now the axes of so y % are perfectly arbitrary, we may there- 
 fore choose them so that the principal axes of the body 
 shall coincide with them at the time , and therefore we 
 shall have at the time #, 
 
 and hence, from the equations just obtained, we have at 
 the time # 
 
 at 
 
 dt dt 
 and hence the first of the equations (A) will become 
 
 ~^m (a? + f) + w^ 2$m (* - y*) = N. 
 
 Now the principal axes coincide with the axes a? y z at 
 the time t, hence if A B C be the principal moments of 
 inertia of the body, we shall have
 
 65 
 
 23m (x* - y 2 ) = 
 
 (y* + *) 
 
 and hence ^. \- (B A) 
 
 AT. 
 
 We may transform the other two of the equations (A) 
 in exactly the same manner, hence, instead of the equations 
 (-4), we have the following three equations, in which the 
 integral sign 2 is got rid of, viz. 
 
 A + (C - B) ft> 2 o> 3 = L 
 at 
 
 at 
 
 70. It remains to determine what w l w 2 w 3 are; the 
 equations (B) will enable us to do this immediately, for 
 from these equations we find (putting % = 0) that the ve- 
 locities parallel to the axes of x and y of any particle in the 
 plane of any are 
 
 daa 
 
 dy 
 
 -= ,,,*; 
 
 and hence the whole resolved velocity in the plane of xy of 
 that particle, viz. 
 
 will be o) 3 <\/a? + y 2 . 
 
 Now if we suppose the particle to be at a distance unity 
 from the origin, and therefore a? 2 + y 2 = 1, this velocity will 
 become a> 3 , and hence w 3 is the resolved velocity in the plane 
 of xy of any particle situated in that plane, at a distance 
 9
 
 66 
 
 unity from the origin. Since the particle is rigidly con- 
 nected with the origin, it is evident that this velocity takes 
 place perpendicularly to the line joining the particle and the 
 
 origin ; and also since is negative, it is evident that this 
 dt 
 
 velocity tends to move the particle from the axis of x to- 
 wards that of y, 
 
 In the same way it may be proved that MI and <o 2 are 
 similar velocities with respect to the planes of yz and zx, 
 respectively ; o>i tending from the axis of y towards that of 
 #, and o> 2 tending from the axis of % towards that of sc. 
 
 71- To give clear ideas, we shall represent the manner 
 in which these velocities tend, by means of the following 
 figure. 
 
 Let XY, FZ, ZX be the intersections of the planes 
 of acy, yz, #, respectively, at the time #, with a sphere 
 fixed in space, unity being its radius, and the origin its 
 center ; then <a 3 will be the resolved velocity along the great 
 circle XY of any point P situated on that great circle, and 
 o>j will be the resolved velocity along the great circle FZ 
 of any point Q situated on that great circle, and o> 2 will 
 be the resolved velocity along the great circle ZX of any 
 point R on that great circle; and these velocities tend in 
 the directions represented by the arrows.
 
 67 
 
 It is evident from this, that wj, o> 2 > >3 are also the 
 angular velocities of the planes of yz, ##, and ay, round 
 the axes of #, y, and *, respectively : but it may be easily 
 seen from the equations (.6), that w lt w 2 5 <Ws are not tne 
 angular velocities round the axes of #, y, ar, of any other 
 points of the body, except those situated in the respective 
 co-ordinate planes. 
 
 72. We may determine the position and motion of the 
 body by means of these velocities, as follows. 
 
 Let A be any fixed point on the surface of the sphere 
 described before (see the figure), and AB any fixed great 
 circle; draw the great circle AZC to meet YX produced 
 in C. 
 
 Let the angle ABZ = \^, the angle A Z = 0, and the 
 angle C Z X = <p; then it is evident that these angles define 
 completely the position of the body in space at the time t, 
 and may be considered as the co-ordinates of the position 
 of the body, \^ and being the co-ordinates of the point Z, 
 which we shall call the pole of the body, and (p the addi- 
 tional co-ordinate requisite to determine the position of the 
 plane of ZX, and therefore of the whole body; we may 
 easily determine these co-ordinates from the quantities 
 !, w 2 , fc> 3 , as follows. 
 
 It is evident that the velocities of the point of the body 
 coinciding with Z are, 
 
 , perpendicular to A Z, 
 o/ 1 
 
 J f\ 
 
 and along ^Z, 
 
 and the velocity along CX of the point of the body co- 
 inciding with C is 
 
 d\l/ 
 
 --| sin AC, due to the variation of >Jr, 
 
 + -T" 5 * ZC* due to the variation of 0;
 
 68 
 
 i. e., since ZC = 90, 
 
 Now by what we have proved in (71), the velocities of 
 the body coinciding with Z are, 
 
 &>2 along ZJT, 
 and - w l along Z F, 
 
 and the velocity along C X of the point coinciding with C 
 is o> 3 ; hence, since these two sets of velocities must be 
 equivalent, we have, resolving the latter set so as to make 
 them coincide with the former, 
 
 -J- sin 9 = MI cos + o> 2 sin < 
 at 
 
 = ojj sin + o) 2 cos 
 
 (C). 
 
 73. These differential equations connect 0, \^, and 6 
 with i, &) 2 o> 3 , and these, along with the three others (A') 
 in (69), which connect o^, w 2 , w 3 with the impressed forces, 
 form a system of six equations connecting the six unknown 
 quantities o) 15 w 2 , ft> 3 , \^, ^, with #, and which therefore, 
 when it is possible to do so, will enable us to solve any 
 problem respecting the motion of a rigid body about its 
 centre of gravity. 
 
 The equations (B) will enable us to determine the mo- 
 tion of any point we please of the body, should it be 
 requisite to do so. 
 
 74. These equations will be sufficient for our present 
 purpose, namely, the determination of the Earth's motion 
 about its centre of gravity ; we shall hereafter recur to 
 this subject, and deduce several interesting consequences 
 from these equations.
 
 CHAPTER VI. 
 
 PRECESSION AND NUTATION. 
 
 75. WE shall now make use of the equations deduced 
 in the last Chapter, to determine the motion of the Earth 
 round its center of gravity. 
 
 The forces which act on the Earth, are the attractions 
 of the Sun, Moon, and other planetary bodies; but on 
 account of the Earth's nearly spherical form, the motions of 
 the Earth round its center of gravity, produced by these 
 forces, are but small ; hence, by the principle of the super- 
 position of small motions, we may consider, separately and 
 by itself, the effect of the attraction of each planetary body 
 on the motion of the Earth round its center of gravity ; we 
 shall accordingly commence with the Sun's effect. 
 
 76. We shall first prove, that the attractions of the 
 Sun on any particle of the Earth are the same very nearly 
 as if the Sun's mass were condensed into his center of 
 gravity. 
 
 Take the center of gravity of the Sun as origin; let 
 xy'% be the co-ordinates of any element m of it, and let 
 xyz be the co-ordinates of the attracted particle m of the 
 Earth. 9 
 
 Then, if V denote the sum of each element of the Sun, 
 divided by its distance from $m, the attractions of the Sun 
 on $m will be 
 
 dV dV dV 
 dx" dy* d%'
 
 70 
 
 \ 
 
 Now V = 2 
 
 2 (o?^ + yy + #*') r' 
 
 putting r and r' for <# 2 + y 2 4- # 2 , and X* 4- y' 2 + #' 2 , re- 
 spectively 
 
 a?<j?' + yy + %% \ 
 
 r } 
 
 expanding and neglecting the squares, &c. of the very small 
 
 x' 11 %' , r 
 
 quantities , , , and - 
 
 r r r r 
 
 1 
 
 = 2owi . 
 r 
 
 Since 2^m,t?', 25my', 2^m^' are zero, because the origin is 
 
 center of gravity. 
 
 Hence F= ; 
 r 
 
 and therefore, neglecting the squares &c. of very small 
 
 dV dV dV 
 quantities, F, and consequently the attractions , , , 
 
 are the same as if the whole mass of the Sun were collected 
 into its center of gravity. 
 
 77- Now take the principal axes of the Earth, passing 
 through its center of gravity, as the axes of co-ordinates, 
 the polar axis being that of % ; let aoy% be the co-ordinates 
 of any element $m of the Earth, and xyss the co-ordi- 
 nates of the Sun, supposed to be condensed into his center 
 of gravity ; then, m being the mass of the Sun, his at- 
 tractions on m, parallel to the axes of x and y, will be 
 
 andF=
 
 71 
 
 and hence the quantity N, (see the equations (A') of the 
 last Chapter) which equals 2$w (Yx Xy)^ will become 
 
 (y'-y)*-(*'->)y \ 
 
 *'-*m^) 2 +(*'-*)i* '"I 
 
 \yoe-aiy) [r' 2 -2(a!X+yy + **') + r 2 ]"! \ , 
 
 r and r being the distances of $m and $m from the origin, 
 and r being therefore very large compared with a?, y\ % 9 
 or r ; hence, expanding and neglecting the squares of very 
 small quantities, we have 
 
 or, since the origin is center of gravity, and therefore 
 2^ma?, S^my, 23m*, each zero, and since the axes are 
 principal axes, and therefore 
 each zero, we have 
 
 and ^f being the same as in the last Chapter. 
 
 Similarly, 
 
 8m' *V 
 
 - Q, 
 
 r 
 
 Sm'y'ss' 
 
 (C-B). 
 
 r' 5 
 Hence the equations (-4')j in the last Chapter, become 
 
 A + (C - B) W2 3 = (C - K)
 
 72 
 
 78. These equations simplify very much in the case 
 of the Earth, for the polar axis being that of *, the 
 moments of inertia round all axes in the plain of xy will 
 be the same, since the Earth is symmetrical with respect 
 to the polar axis ; hence B = A, and the last equation will 
 become 
 
 and therefore o> 3 = constant = n suppose. 
 
 That is to say, by (71), the angular velocity of the plane 
 of the equator round the fixed axis with which the polar 
 axis coincides at any time, is a constant quantity. 
 
 Now if ri be the Sun's mean angular velocity relative to 
 the Earth, 
 
 n' 2 = , m being the Earth's mass, 
 
 = , very nearly, m being very small com- 
 
 pared with TO'; 
 
 C A 
 hence, and putting ft for - , the other two equations 
 
 become 
 
 79- Now in the figure (page 66), let S be the point 
 where a line drawn from the center of the Earth to the Sun 
 meets the fixed sphere mentioned in page 66; draw the 
 great circle SZ, then SZ will be the Sun's north polar dis- 
 tance, which we shall denote by A, and SZX will be the 
 Sun's hour-angle relative to the meridian plane ZX, which 
 we shall denote by h : it is evident that A and h are the 
 polar co-ordinates of the Sun, and therefore we have
 
 73 
 
 x = r sin A cos A, 
 y = r sin A sin A, 
 
 z' = r cos A. 
 
 hence -r- = sin A cos A sin A, 
 r a 
 
 ' = sin A cos A cos h ; 
 r a 
 
 and hence our equations become 
 
 + w/3 w a = 3w' 2 /3 sin A cos A sin A, 
 
 2 - w/3o>i= 3n' 2 fi sin A cos A cos A. 
 ct 
 
 We shall take a year as the unit of time, and hence n', 
 
 2ir 
 
 which = - , will become 27r, and n will be about 365, 
 a year 
 
 also we shall shew that /3 is about = . 
 
 330 
 
 Now h varies in consequence of the Earth's diurnal 
 rotation, and also in consequence of the Sun's motion and 
 the motion of the polar axis ; but the part of its variation 
 due to the former cause is very much greater than that due 
 to the latter causes, hence we may put 
 
 dh 
 
 where $n is a small quantity compared with n, depending 
 on the motions of the Sun and polar axis, we put - n because 
 h decreases with the time. 
 
 Hence in the first of our equations, putting 
 d<a\ dh)\ dh dto, , * 
 
 it becomes 
 
 da,, nfi 3n'*fi 
 
 e> 2 = s sin A cos A sin n ; 
 
 dh - n 4- $n - n + 
 10
 
 74 
 or, neglecting $n compared with w, 
 
 - - /3ft> 2 = Sra'* sin A cos A sin h. 
 ah n 
 
 Now the second member of this equation is very much 
 smaller than the first, on account of being multiplied by 
 
 - , hence in it we may, when integrating, consider the 
 n 
 
 periodical quantity A as invariable, since it varies very 
 slowly compared with h ; and therefore putting, for brevity, 
 
 3n' 2 ft . 
 
 sin A cos A = y, 
 n 
 
 our equation becomes 
 
 ^ -, -y sin h... (1), 
 
 where y is a very small quantity which may be considered 
 invariable in integrating. 
 
 In the same way we shall have 
 
 -~- + 
 ah 
 
 80. Now differentiating (l), and adding (2) multiplied 
 by ft to it, we have 
 
 _J2 
 
 -jj + ft 2 u, } = (- y + 7 /3) cos h 
 
 = - y cos h, 
 neglecting ft compared with unity. 
 
 The integral of this equation will be 
 
 o>i = A cos (fih + B) + C cos h, 
 
 where A and B are arbitrary constants, and C a constant to 
 be determined by substitution. Now A and B, since they 
 depend only on the initial circumstances of the motion, are 
 independent of the Sun's action ; and, as it is our object to
 
 75 
 
 determine the effect of that action alone, we shall omit the 
 term 
 
 Acos(fih+ B), 
 
 which does not depend on it, and we shall have simply 
 
 ( D l = C cos h, 
 
 so far as the Sun's action is concerned ; substituting this 
 value in the equation, we find 
 
 C(-i + /r> = - 7 , 
 or C = y, neglecting ft 2 ; hence 
 
 fcjj = y cos h. 
 
 In like manner we shall have, differentiating the equa- 
 tion (2), and subtracting (1) multiplied by /3 from it, 
 
 d 2 a)2 . 
 
 + /5 2 ft> 2 = - 7 sin h ; 
 an 
 
 and therefore, as before, 
 
 ft> 2 = y sin h. 
 
 81. Having thus determined &>! and ft> 25 we shall sub- 
 stitute their values in the equations (C) (last Chapter), in 
 order to determine 9 and 0, and so find the position and 
 motion of the pole ; we have then, substituting the values of 
 o>i and ft> 2 just obtained in the two first of the equations (C), 
 
 dvl/ 
 
 ~ sin 9 = y sin h sin - y cos h cos <p 
 
 = - y cos (h + 0), 
 
 dO 
 
 and = y sin h cos (f> + 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 
 
 <ty 3n' 2 
 I_ = r cos sm 2 /, 
 
 dt n 
 
 d9 3n'*P . 
 
 a- _ _ L- SJn C os /. 
 
 dt n
 
 77 
 
 83. Now being very small, we may in integrating 
 n 
 
 these equations consider & invariable in the terms multiplied 
 
 by ; and we may also, for the same reason, consider I to 
 
 n dl 
 
 vary uniformly, and therefore put = n ; hence, putting 
 
 d\/ d\, dl d 
 
 d9 dO , 
 
 Tt = -di n > 
 
 and sin 2 / = 1 (l - cos 2 Z), 
 our equations become 
 
 S * 
 
 hence, integrating 
 
 ^ + C = - - - y3 . cos 6 (I - J sin 2/), 
 
 Q + C" = 3 - 3 sin sin /. 
 n ^ 
 
 In the second members of these equations may be 
 considered as the mean obliquity of the ecliptic, and may 
 therefore be determined by astronomical observations; and 
 
 , the ratio of a year to a day, may be similarly deter- 
 n 
 
 mined ; and thus we may put our equations in the form 
 
 9 + C = gfi sin /, 
 
 where e', f ', g', are numerical quantities got from obser- 
 vation. 
 
 These equations determine the effect of the Sun's at- 
 traction on the motion of the Earth round its center of 
 gravity.
 
 78 
 
 84. To determine the effect of the Moon on the Earth's 
 motion round its center of gravity, we may proceed in ex- 
 actly a similar way, merely supposing the symbols which 
 before referred to the Sun now refer to the Moon, and alter- 
 ing them accordingly, as follows : 
 
 Let m y , w /5 and r t be the mass, mean angular velocity, 
 and distance of the Moon, then as in Art. (78), we shall 
 have 
 
 here m instead of being much smaller than m t as before, is 
 much larger than it, and therefore cannot be neglected 
 as before, 
 
 let X = , then 
 
 n 
 
 m 
 
 , 
 
 and therefore ' = - ; 
 r, 3 X + 1 
 
 hence, in changing our equations so as to refer to the Moon, 
 
 M* 
 
 we must in substituting for n 2 put ' instead of nf 
 
 for it. 
 
 Hence, if we put dashes under the letters to denote 
 that they refer to the Moon, we shall have for the Moon's 
 effect on the Earth's motion round its center of gravity, 
 
 s ' / - sin '"' 
 
 Of course the point A in the figure, (page 66), is now 
 supposed to be the pole of the Moon's orbit, and not the 
 pole of the ecliptic.
 
 79 
 
 85. In these expressions the coefficients are much 
 smaller than before; so small that the periodical quantities 
 multiplied by them, viz. sin 2n t t and sin nt^ which go 
 through all their values in half a month, and a month 
 respectively, may be neglected, this will give 
 
 and 
 
 and therefore --- = - - - cos 9 , 
 dt 2n (X + 1) 
 
 , dB 
 
 and ' = 0, 
 dt 
 
 Which equations prove, that the effect of the Moon's 
 attraction (omitting very small periodical quantities of short 
 period) is to produce a motion of the pole of the Earth per- 
 pendicular to the great circle AZ, i.e., the great circle 
 joining the pole of the Earth and the pole of the Moon's 
 orbit ; and the velocity with which this motion takes place 
 
 (i.e. sin 0,), is 
 
 cos 9 sin 
 
 We shall resolve this velocity along and perpendicular 
 to the great circle, joining the pole of the Earth and the 
 pole of the ecliptic, in order to get our quantities measured 
 in the same way as before in the case of the Sun, and so de- 
 termine the variations (due to the Moon's action), of the 
 angles \js and 0, which refer to the pole of the ecliptic. 
 
 86. Let i be the inclination of the Moon's orbit to the 
 plane of the ecliptic ; then it is evident that t, 0, and 9 t form 
 the sides of a spherical triangle.
 
 80 
 
 Let cr be the angle made by 6 and /? then resolving the 
 velocity 
 
 3 2 /3 
 
 cos 9 sm 6 , 
 2w (X + 1) 
 
 (which acts perpendicularly to y ), along and perpendicular 
 to 9 we find for the resolved parts, 
 
 -- 7-* - - cos 9 sin cos a perpendicular to 0, 
 2n (\ + 1) 
 
 and -- -r-* - - cos 9 f sin 9 t sin cr along 0, 
 2 w (A. + 1} 
 
 and hence, since sin 9 and are these velocities, we 
 at at 
 
 have the following equations to determine the effect of the 
 Moon's action on \!/ and 9, viz. 
 
 d\js 3nffl cos Q t sin 9 f cos cr 
 
 1/7 2n(\+ 1) sin 
 
 d0 3w//3 
 
 = -- cos 9 sin sin cr. 
 
 dt 
 
 Now in the triangle of which t, 0, and / are the sides, 
 o- is the angle opposite i, and if & be the longitude of the 
 Moon's node, it is evident that & is the angle opposite B t ; 
 hence we have 
 
 cos t = cos 9 cos / + sin sin y cos cr, ... (l) 
 cos Q t = cos i cos + sin i sin cos & , ... (2) 
 sin 0. sin t 
 
 (3) 
 
 am 56 alu O" 
 
 (n rt sin A r-i\< rr 
 
 hence 
 
 sin0 
 
 COS t - COS COS 0,
 
 = (cosi cosfl + siru sin0cos& ) 
 
 81 
 
 cos t sin 2 sin t cos0 sin 9 cos & 
 
 sin 2 9 
 
 (sin 2 0-cos 2 0) 
 
 cos 2 i cos 0+sm t cos i - - - cos & -sm 2 t cos 9 cos 2 & 
 sin 9 
 
 : 
 sin 9 
 
 and also 
 
 cos 9, sin 9 1 sin a- 
 
 = (cos i cos 9 + sin t sin 9 cos & ) sin t sin 
 = ^ sin 2t cos 9 sin & + ^ sin 8 1 sin sin 2 & . 
 
 In these expressions Q> alone may be considered as va- 
 riable, all the other quantities varying very slowly, and 
 within very small limits; also, since these expressions are 
 to be multiplied by a very small coefficient, we may, in the 
 periodical terms, neglect sin 2 t, since t is not much more 
 than 5. 
 
 Also, if v be the mean angular retrograde velocity of 
 the Moon's nodes, we may put 
 
 and hence we shall have as in former cases 
 
 3w 2 /3 A sin 2 * 1 
 
 ' cos0{cos-< ----- sin 2 1 cot 29 cos Q }, 
 
 i/(\+l) 1 2 j 
 
 cos0sm2t sin & ; 
 
 - -- 
 d& 2ni/(\ 
 
 d9 
 
 and hence, integrating, 
 
 9 + C' = - - / ^ ^ cos0 sin2t cos & ; 
 
 11
 
 82 
 
 or, as before, we may put these equations in the form 
 
 3 3 
 
 ^ t c-^ -/ 
 
 A + 1 
 
 When e^ f t , and g are numerical quantities, got by ob- 
 servation. 
 
 Thus we have determined the effect of the Moon's 
 attraction on the motion of the Earth, round its center 
 of gravity. 
 
 87- The effects of the other planetary bodies are very 
 small indeed, and we shall neglect them ; hence, adding 
 together the effects of the Sun and Moon, we find for the 
 whole motion of the Earth round its center of gravity, so 
 far as it is affected by external attractions, 
 
 >/, + C = - /3 (el - ~^~ Q ) + /'/3 sin 2 / 
 
 A. + 1 
 
 0+ C' = 's\i 
 
 / 
 A + I 
 
 The first term of the expression for >// + C is non- 
 periodical, and its rate of variation is 
 
 or -(3(e'n' + ^-_ 
 
 which being constant and negative represents a uniform 
 retrograde motion of the pole in longitude : it gives rise to 
 what is called the precession of the equinoxes, because, in 
 consequence of it, the first point of Aries moves constantly 
 backwards, and therefore the equinox occurs sooner than it 
 otherwise would every year.
 
 83 
 
 Observation shews that this retrograde motion of the 
 pole in longitude is about 50".l per year; hence we ought 
 to have 
 
 The other terras of \|/ + C and + C' are periodical, 
 depending on the longitude of the Sun and of the Moon's 
 nodes; they are called the solar and lunar nutations. Ob- 
 servation shews that the coefficient of sin & is about 18", 
 and that of cos Q about 9".5, the coefficients of the other 
 terms are much smaller ; hence we ought to have 
 
 and 9.6-r,r-- ............ (3). 
 
 A + 1 
 
 88. Since e , / /5 and g are known numerical quan- 
 tities, it is evident, that from (l) combined with (2) or (3), 
 we may eliminate )3 and find X ; the result is 
 
 X = about 70, 
 
 thus by observation on precession and nutation we may 
 determine X, which is the ratio of the Earth's mass to that 
 of the Moon. 
 
 By the same equations we may determine /3, the re- 
 sult is 
 
 /3 = about .00319, or - . 
 *oO 
 
 89- Now /3 may be also calculated by integration, if 
 we know the law of arrangement of the Earth's mass ; we 
 shall calculate /3, assuming the results arrived at in Chap, in, 
 and if we find that the value of /3 thus obtained coincides 
 with that just determined by observation, it is evident that 
 we shall have an additional proof of the correctness of our 
 hypotheses in Chap. in.
 
 84 
 
 90. We have evidently 
 
 -) 2Sm (*-) 
 
 ' S1 
 
 #, y, # being the co-ordinates of any particle ($m) of the 
 Earth, the polar axis being the axis of ss ; or using the polar 
 co-ordinates as before, 
 
 Now putting ^ + ^ cos 20 for cos 2 0, and integrating 
 relatively to 0, observing that r does not contain 0, the 
 numerator of )3 becomes 
 
 which, putting 
 
 r'= 5 {l + 5e(J -!.")}, 
 
 and observing the property of Laplace^s coefficients in Art. 
 21, becomes 
 
 9 9 - 
 
 8 
 
 ~ 45' 
 
 hence our integral becomes 
 
 Sir ra d(o 5 e) . 
 / p -- da. 
 
 15 Jn 
 
 da
 
 85 
 In like manner the denominator of /3 becomes 
 
 TT f r, d(O/4 /i \1 
 = - / / p : { -- -- M V 
 5 J-\ Jo r da (3 \3 ) I 
 
 Since, on account of the smallness of /3, we may neglect e 
 in its denominator. 
 
 Hence, observing the property in Art. 21, the denomi 
 nator of /3 becomes 
 
 STT /<! rf(a 5 ) STT /. 
 
 / p aa which = / pa*da. 
 
 15 Jo * da 3 A ^ 
 
 hence we have 
 
 5 
 
 / p 1 
 
 1 <J Q (t (t 
 
 r^ 
 
 I pa*da 
 
 /o 
 
 Now the equation for determining e in Art. 53, gives 
 
 e, /"' . 1 r a ' d(a 5 e) a> z a j 
 
 -- ^ / pa'da + - / p da + -- ^ = 0, 
 a,J r */' ^a STT 
 
 r "> w^o/ 
 
 a e / pa^da -- '- 
 
 ' Vt r STT 
 
 hence p = - 
 
 rpa* da 
 . ' 
 
 a~ I pa? da 
 
 ' Jo ' 
 
 I pa*da
 
 putting 
 
 m being, as before, the ratio of the centrifugal force to 
 gravity at the equator. We can go no farther in calculating 
 /3 without knowing the law of density ; hence, taking the 
 law already assumed, and substituting in the integrals and 
 performing the integrations, we shall find 
 
 /3 = about .0031359, or . 
 
 330 
 
 91. Hence this value of /3 coincides with that got from 
 observation, in Art. 88, and we have therefore an additional 
 proof of the hypothesis of the Earth's fluidity, or rather, 
 of the assumed law of density ; for since this result cannot 
 be obtained without assuming the law of density, it is not of 
 much value in proving the hypothesis of original fluidity ; 
 but we may consider that hypothesis as well established by 
 previous results, and then the coincidence of the values of /3 
 will go to proving the probability of the assumed law of 
 density. 
 
 END OF PART I.
 
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 12 Theology. 
 
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 Theology. 13 
 
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 Tlieology. 15 
 
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 THE THIRTY-NINE ARTICLES, TeJlimonies 
 
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 LEXICON to the NEW TESTAMENT, a Greek 
 
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 BP. PEARSON'S FIVE LECTURES on the Acts 
 
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 Classics. 21 
 
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 Miscellaneous Works. 23 
 
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 Mr. MACAULAY'S Character of the CLERGY 
 
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 REMARKS on the ARCHITECTURE of the 
 
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 A 000426824 9
 
 
 
 Sh'n'tlii n-ilt hs PuUixlicd. !>;! the wwr Aitthnr : 
 
 MATHEMATICAL TRACTS, 
 
 PART II. 
 
 Containing, among other subjects, The MATHEMATICAL 
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