HOMOGENEOUS FLUID AT LIBERTY. 511 



the upper surface ; because by the nature of the functions C^*^ all the integrals vanish 

 when R' is constant. But the supposition of a sphere requires that g be equal to zero 

 in the equations (1.) and (2.), or that there be no centrifugal force. 



But if the radius R vary as it changes its direction, / —r cannot be of the same 



quantity at every point of the same level surface, except when all the terms after the 

 first are separately equal to zero, that is, except the expression of R' be such that 



ff:^^^^ = o, 



for all values of i from 1 to x . 

 The investigation will be greatly facilitated by the following theorem : 

 If a' = cos &, V = sin & cos z?/, c' = sin & sin -us^ the integral 



ff-dydcd-'^arV'c'^', 



extended to all values of y from 1 to — 1, and of a from to 2 cr, will be equal to 

 zero in all cases when w + m' + m" is less than i*. 



It is obvious that -p/a is a function of the three quantities a!, h', d ; and if we , 



assume 



J^ = u'"' + u« + u«, 



U being a constant, and U^ , U^^ functions such that a!, V, c' rise to one dimension 



in all the terms of U^^ , and to two dimensions in all the terms of U^ , the highest 

 sum of the indexes in the combinations of a!, V, c', contained in the expressions of 



R^' R°"' R?^ ^^'^ ^^^^ ^^^ exceed 4, 6, 8, &c. : wherefore, by the theorem, the 



1 f* dm . , 



assumed value of ^ will succeed in making all the terms oij —j- vanish in which 



1 



i is an even number, and it is evidently the most general assumption for -^ that will 



answer the same end. 

 When I is an odd number, we have 



f P -dyd(TC^^ _ f f -dyd<rC^i) 



In this case C^' , being an odd function of y, is the same in quantity, but changes 

 its sign, when for ^ and vr' we substitute ^ + -^ and w' + ?r : wherefore the whole in- 

 tegral will be equal to zero, if the denominator retain the same positive value when 

 0' and tst' are changed into ^' + -^ and tj' + t, the increase of the integral being, on 



* Vide Appendix. 

 3 u2 



