r 



contained in the Third Book of the Mecanique Celeste. 435 



discussion, it may be observed here that, in what follows, the 

 value of the integral is not estimated by the ordinary rules. 

 The reasoning turns on this principle : that whatever is proved 

 of every individual element or differential must be true of the 

 aggregate or integral. No principle certainly can be more 

 clear or less exceptionable ; and all that we have to do is to 

 apply it strictly, and to assure ourselves that in every case the 

 property to be proved does really belong to every individual 

 element without exception. 



In the first place, when the exponent n is positive and greater 

 than 1, the equation at the surface is true in the most exten- 

 sive sense : for every element of the integral in the second 

 formula being finite, any aggregate of them will be equal to 

 nothing on account of the evanescent factor. 



We come next to the case when n is negative. If we write 

 — 71 for 71, the second formula ( 1 ) will become 



- (r^-g") {y' -y)ds ; 

 ^„ + l 



and here we see that when cos\I/ = 1, andy = r — a, the ele- 

 ment of the integral may, on account of the denominator, be- 

 come infinitely great, notwithstanding the evanescent factor in 

 the numerator. But in order to investigate this case with 

 clearness, there are some considerations to be attended to. 



It is to be observed thatj/' — y = O when cos \I/ = 1. As 

 the molecule approaches the point of contact of the sphere and 

 spheroid, we may suppose that its thickness decreases in the 

 same proportion withy'^, the square of its distance from that 

 point. It appears from the explanation which the author has 

 given in his later writings *, that this circumstance is to be 

 understood in the demonstration in the Mecanique Celeste; for 

 it is not expressly mentioned, and no stress is laid upon it. It 

 was the more necessary to be explicit on this head, because the 

 property assumed may not belong to every function thaty may 

 be supposed to stand for. It seems an omission not to distin- 

 guish the cases that come under the demonstration from those 

 to which it will not apply. It is curious too that this point es- 

 caped the penetration of Lagrange, whom it would have helped 

 to clear up that analytical mystery which he calls a paradox 

 in the integral calculus f. 



Another thing to be observed is, thatj/' —y being consi- 

 dered as a function of vj/ and ip, the element of the surface of 

 the sphere ds will be ecjual to a^ d^i sin vf/ rf (j5; whence it fol- 

 lows that near the point of contact ds will decrease in the 



• Mecanique Celeste, livre 11"", chap. 3. 

 f Journ. dc VEcol. Vulyt. torn. 8. 



3 I 2 same 



