We have here followed very closely all the steps of the demonstration contained in the Mécanique Céleste, and on first thoughts no reasoning can be more convincing, or appear more free from all obscurities. This much at least is certain, that every part of the demonstration is placed beyond the reach of all objections except the valuing of that term in the equation (B), which is derived from the difference between the spheroid and the sphere: and about this a deeper consideration of the nature of the functions concerned may raise in the mind some doubts and scruples. No better way can be devised for trying the soundness of LAPLACE's procedure, than to perform that part of the calculation which is alone liable to suspicion, without omitting any of the terms which he has tacitly rejected; to throw out such only as on examination can be proved to be necessarily evanescent when dr=0; and to retain the rest if there be any of a different description. Now, to apply this rule, we have f2 = 2p3 (1—y); and ƒ”= (p + dr)• − 2p (p + dr) y + p2 = { 1 + *} . 2p′ ( 1−y) + ♪r"; therefore ƒ” — d' = { 1+ #} .ƒ': consequently,=

f12

{1+ *}* × {1}; and, by expanding the second radical into a

and, by multiplying by dm and affixing the sign of integra

This expression being farther reduced into a series of simple terms, those terms will be included either in the form

dm

or in the form didm whatever number i

:

fi+1 may denote, the first sort of terms, when they are integrated between the proper limits, will be found, on examination, to contain a part which, depending only on the nature of the molecules or of the function that expresses the thickness of the molecules, remains of the same magnitude for all values of dr; and consequently those terms do not necessarily vanish when dro: with respect to the second kind of terms, they are to be regarded as quantities of the same order with the multipliers written without the sign of integration, and they all vanish together with dr. Reserving till afterwards the proof of what has now been said, it is sufficient at present to have marked distinctly the characters of the quantities to be retained, and of those to be rejected. If then we retain the first sort of terms only and reject the rest, the value of

and, by substituting this in the equation (B), we shall get

in which expression the value of all the terms under the sign of integration are to be taken on the supposition of dr = 0. Finally if, in this last value of () and the value of V already

dV
dr

found, we first substitute a . (1 + a . y) for p, retaining only the quantities of the first order with regard to a; and then combine the two expressions so as to exterminated, we shall get the following equation instead of that of Laplace,

2. In order to find the integrals in the equation (C), we must begin with seeking an analytical expression for the value of dm, which may be conceived to be a prism standing on an indefinitely small portion of the spherical surface, and limited in its height by the surface of the spheroid. Let p' denote the radius of the spheroid drawn to the molecule dm, and 6′ and 'the angles which determine the position of p' in like manner as 6 and a determine the position of p: and, if y' be put for the same function of ' and 'that y is of 0 and w, then p'= a. (1+a.y'). Suppose 6' and ', the arcs which determine the position of p', to vary; and the correspondent fluxion of the spherical surface whose radius is p, will be = p. sin. 6′. de'. do' = (being put for cos. 6') p. du'. da'; this is the base of the prism equal to dm: the height of the prism is plainly p' p= a. a. (y'-y): therefore dma.a.

p2. (y' —y).

du'. da': and, by substitution, the equation (C) will become

v+a.

Since r,

the distance of the attracted point from the centre,

is = ç + dr, and ƒ' = {r2 — 2rp. y+p'}; therefore the ge

neral term of the series in the last equation will be

and because y' is a function of the variable angles é′ and ✩', or of μ' and '; and y is a constant quantity; therefore, if u' be put to denote a function of the angles ' and ', both the integrals in the general term will be obtained by investigating the integral

for the whole surface of the sphere, and in the particular circumstance of r=p, or r— p = 0.

3. The formula which is now to be considered cannot be integrated without limiting the symbol u' to denote a particular function, or class of functions. But LAPLACE's demonstration will be completely overturned, if it shall be shown that, in any hypothesis for u', the formula in question has a finite value when rpo: for then the only reason which he can be supposed to assign for rejecting such terms in the value of (a); namely, that they contain a vanishing factor, must be allowed to be inconclusive. We shall henceforth suppose that v' denotes a rational and integral function of μ', ✔I— μ22. cos. ', √1—μ3. sin. ', which are three rectangular co-ordinates of a point in the surface of a sphere; a supposition which in effect embraces the whole extent of LAPLACE'S method.

dr

The demonstration which LAPLACE has given of his fundamental theorem is independent on the function y, being drawn entirely from the nature of the algebraic expression of the distance between the attracted point and a molecule of the

و سلم

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matter spread over the surface of the sphere.* From this circumstance indeed is derived one great advantage of his method, namely its great generality; for no restriction whatever is imposed on the nature of the spheroid excepting that of a near approach to the spherical figure. Nevertheless the author, by means of a simple transformation, immediately deduces from his theorem an equation which proves that y and V are expressed by two series both containing the same sort of terms: and since all the terms of the series for V can only be rational and integral functions of cos., 1μ. sin. ; it follows that y must be a like function of the same three quantities. We may remark here that this consequence of LAPLACE's reasoning appears to be inconsistent with the premises: for it is hard to reconcile with the rules of legitimate deduction that an equation obtained by supposing y to be arbitrary, should, merely by having its form changed, be made to prove that the same quantity must be restricted to signify a function of a particular kind. But we mention this only by the bye, without meaning to insist upon it; although we cannot help thinking that it ought to have led the learned author to entertain suspicions of the accuracy of his calculations; all that we intend by the foregoing observation is to prove that in point of fact we shall embrace the whole extent of LAPLACE'S method by supposing y to be a rational and integral function of three rectangular co-ordinates of a point in the surface of a sphere.

Supposing then ' to denote such a function as has been mentioned, we are to investigate the value of this integral, 1 Liv. 3e, No. 9.

* Liv. 3e, No. 10. MDCCCXII.

+ Liv. 3e, No. 11.

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