contained in the Third Book of the Mecanique Celeste. 437 



whole surface of the sphere, or covers it in any manner par- 

 tially. It remains to ascertain in what cases the supposition 

 assumed is true, an inquir}' totally omitted by the author of 

 the demonstration. It must be recollected that j/ is a func- 

 tion of the arcs fl' and ct' ; that y is the value of j/' when V and 

 ts' have the particular values 9 and -ot; and that \^ is the arc 

 on the surface of the sphere between y and y. When r ■= a, 

 theny^ = 2 a- {I — cos v}/) ; and the supposition on which the 



demonstration proceeds is evidently that ^'^^of ' 'Tiust have 



a finite value when the numerator and the denominator are 

 both evanescent. We want to know to what class of func- 

 tions this property belongs ; for to such only the demonstra- 

 tion will apply. If it were asserted that the demonstration is 

 exact, whatever function y' stands for, it would be sufficient 

 for destroying it, to show that it fails in one particular mstance. 

 But, in order to avoid minute discussion, we may remark that 

 the analysis of Laplace is solely employed about rational and 

 integral functions of three rectangular co-ordinates of a point 

 in the surface of a sphere ; and we shall prove that such ex- 

 pressions do not come imder the foregoing demonstration. — 

 Now if y be a function of the co-ordinates, cos 5', sin S' cos 'bt', 

 sin 9' sin ct', it may, as already noticed, be transformed into 

 a similar function of the co-ordinates cos ^, sin 4/ cos <p, sin \|/ 

 sin (p : and again, by substituting for the powers and pro- 

 ducts of sin (p and cos y, their values in the sines and cosines 

 of the multiples of the arcs, the latter function will take this 

 form ; viz. 



H(o\ + Hti^sin^l/cos^ + &c. 

 + K'^' sua ^ sin <p + &c. 

 the symbols H^"', H^^', K'^', standing for rational and inte- 

 gral functions of cos ^^. Wherefore y' — y will be equal to 

 (Ht"'— j/) + Ht^)sin\J/cos<p + &c. 

 + K'^~ sin \{/ sin ip + &c. 

 In this expression all the terms vanish separately when\{> = 0; 

 whence it follows that H '^' — j/ is divisible by 1 — cos ^. But 

 the two next terms, multiplied by cos p and sin a, are not so 

 divisible, the quotients in both cases being infinitely great, 

 instead of being evanescent, when cos v{/ = 1. Wherefore the 



quantity y^,^-;^ does not tend to a finite limit, but is ulti- 

 mately infinitely great ; and the demonstration entii'ely fails in 

 the particular case considered, which is in reality the only one 

 for which it is wanted. 



W'e know however that, when y' is a rational and integi"al 

 function of three rectangular co-ordinates, the equation at the 



surHxce, 



