[ 103 ] 



XIII. Note on the Integral fdiK-r- \/(m — x)(x + a)(x-f-b)(x + c). 

 By A. Cayley, Esq.* 



IF in the formulae of my t( Note on the Porism of the in-and- 

 circumscribed Polygon," it is assumed that 



U=a? 2 + y 2 + s 2 + -i- (ax 2 + by 2 + cz 2 ) 



V=az 2 + by* + cz 2 , 



and if a new parameter co connected with the parameter w by 



the equation 



com 



w = 



m — co 



be made use of instead of w, then 



W \J + Y= -^~-{co(x* + y 2 + z 9 )+awZ + by* + cz*} . 



or the equation w;U + V = 0, viz. the equation 



co{x* + y 2 + z q ) + ax 2 + by 2 + cz 2 = 0, 



is precisely of the same form as that considered in my 

 Note on the Geometrical Representation of the Integral 



J dx -4- v (x + a) (x -f b) (x + c) . Moreover, introducing instead of 

 f a quantity rj, such that 



m-—if 

 then 



d% __ a/»i drj 



VUZ */{m—<r))(a + 7))(b + 7))(c + r)) 

 Also f =00 gives 77 =m, the integral to be considered is therefore 



*S mdi) 





V [m — rj) {a + rj) (b + 7))(c + 17) 



*. e. if in the paper last referred to the parameter x had been 

 throughout replaced by the parameter m, the integral 



dm) 



n v =f 



co \Z(a-f97)(6 + ?7)(c + ?7) 



would have had to be replaced by the integral II/7. It is, I 

 think, worth while to reproduce for this more general case a 

 portion of the investigations of the paper in question, for the 

 sake of exhibiting the rational and integral form of the alge- 



* Communicated by the Author. 



