the Integral fdiX-T- -/ (x + a) (x + b) (x + c) .1 283 



then the algebraical equations (*) are equivalent to the trans- 

 cendental equation 



+n(^)±n(jo)+n6'=0; 



the arbitrary constant which should have formed the second 

 side of the equation having been determined by observing that 

 the algebraical equation gives for p = 6,k—co a system of values, 

 which, when the signs are properly chosen, satisfy the transcen- 

 dental equation. In fact, arranging the rational algebraical 

 equation according to the powers of k, it becomes 



k^{p-ef (*) 



-U{pe{p + e)-\-2{a-\-b-\-c)pd-{-{bc-\-ca'\-ab){p + e)-\-2ahc} 

 -\-pW^-2{hc + c« + ab)pe'^4abc{p + 6) 

 + a^^b^^c'^-2bc-2ca-2ab=0; 



which proves the property in question, and is besides a very con- 

 venient form of the algebraical integral. The ambiguous signs 

 in the transcendental integral are not of course arbitrary (indeed 

 it has just been assumed that for p==6, Up and 11^ are to be 

 taken with opposite signs), but the discussion of the proper 

 values to be given to the ambiguous signs would be at all events 

 tedious, and must be passed over for the present. 



It is proper to remark, that 6=p gives not only, as above 

 supposed, ^ = co, but another value of k, which, however, cor- 

 responds to the transcendental equation 



±Uk±2Up=0. 



The value in question is obviously 



^j_ p'^-2{bc-\-ca + ab)p^-Sabq)-[.a^+b^^c^-2bc-2ca-2ab 



(p + a)(p-hb)(p + c) 



Consider, in general, a cubic function a^^ + Shx^y + Scxy'^ + dy^, 

 or, as I now write it in the theory of invariants, (a, b, c, d) (^, yY > 

 the Hessian of this function is 



(ac-b^ I (ad-bc), bd-c^) (.r, y)% 



and applying this formula to the function {p + «) ( jo + b) [p -\- c)j 

 it is easy to write the equation last preceding in the form 



.T / .1. X 9 Hessian (» + «» + Z>» + c) 



4>k=p — (a-^b + c) 7 ^ ^ — ^-r — -, 



^ ^ ^ ^ ^ {p + a){p^b){p-\-c) 



which is a formula for the duplication of the transcendent Ux, 

 Reverting now to the general transcendental equation 



+n(A;)±n(jo)±n(^)=o, 



U2 



