436 Mr. Ivory on the TJiewy of the Figure of the Planets 



same proportion with f, and the quotient of the former, di- 

 vided by the latter, will tend to a finite limit. 



These things being premised, the foregoing integral may be 

 thus written, 



and according to what has been said, the two factors on the 

 right hand have always a finite value ; but when cos \l/ = 1, and 

 f =. r — a, the remaining factor, instead of being evanescent, is 

 finite when n = 3, and infinitely great when n is greater than 

 3. In both cases the demonstration fails : in die one, at least 

 some further discussion is necessary ; and in the other, the ele- 

 ment of the integral, and consequently the integral itself, are 

 both infinitely great, instead of bemg evanescent. The equa- 

 tion at the surtace of the spheroid must therefore be under- 

 stood with some restriction when the attractive force is sup- 

 posed to be proportional to any power of the distance. It is 

 true for all positive powers ; but it holds not when the expo- 

 nent of the power is negative and greater than 3, at least when 

 the stratum of matter is spread over the whole surface of the 

 sphere. In his later writings the author makes no mention of 

 the general demonstration he had given in the Mecanique Ce- 

 leste; he confines his views to the law of attraction that ob- 

 tains in nature, which is certainly the most important case ot 

 all, as it is the only useful one. We have next to examine 

 this case particularly. 



In the case of nature, when the attraction is inversely pro- 

 portional to the square of the distance, the quantity A is the 

 sum of the molecules of the sphere divided by their distances 

 from a point without the surface ; it is therefore equal to 



: and hence, ?• — - = : whererore, since ?i = — 2, 



the formulae (1) will become 



^ ' dr 3 I 



. ^ > (2) 



Now, admitting thatj/' — y decreases near the point of con- 

 tact in the same proportion withy-,* it is evident that the se- 

 cond formula will be always equal to zero, on account of the 

 evanescent factor. The equation at the surface of the sphe- 

 roid is therefore, on this hypothesis, strictly demonstrated, 

 and is true whether the stratum of matter is spread over the 



* Mecanique Cckste, livre 11"". 



whole 



