458 Mr. G. B. Jerrard on some Objections 



for n= 6, m=5*, that u and v designate two rational six- valued 

 homogeneous functions of the roots of the equation 



^ 5 +'A 1 a? 4 +A 2 a? 3 + .. + A 5 =0; 



then by Lagrange's theoiy we can generally express either of 

 these functions in rational terms of the other and of the coeffi- 

 cients Aj, A 2 , . . A 5 . The only exception, according to that theory, 

 is when there are equal roots in (U) or (V), the equations on 

 which u y v respectively depend, — or rather when the number of 

 unequal roots in (U) differs from that of unequal roots in (V). 



ii. " But I maintain that when u and v are taken equal to 

 those particular six-valued functions represented by 



,5 



a, a° 



the theory of Lagrange will afford us no aid whatever in esta- 

 blishing a rational communication between them, although in 

 this case the equations (U), (V) have undoubtedly no equal roots. 

 Vain, I assert, must every attempt be to proceed beyond 



a 5 =(a) 5 . 



iii. " That there is something exceptional in the case in ques- 

 tion, we may see at once without entering into the calculations. 

 " For beside the equation 



* J =/*5+A t 4 w + /% w2 + ••+/*(> ^ 5 > • • • ( e i) 



for expressing v in rational terms of u, — which equation in ordi- 

 nary cases would, if there were no equal roots in (U), (V), be 

 sufficient for the purpose intended, — we have, in the present 

 instance, 



v=u 5 (Ej) 



Does (Ej), I would ask, exert no disturbing influence on the 

 coefficients of (e x ) ? 



' ' Again, in ordinary cases we can generally obtain a single 

 definite equation of the form 



uz=v 5 +v 4 v-\-v 3 v*+..+v v 5 ; . . • (*? 2 ) 

 but here we set out with supposing that 



u=^/v; (E 2 ) 



that is to say, u is to be both a rational and an irrational func- 

 tion of v. 



"What wonder, therefore, if, in a case so anomalous, La- 

 grange's theory should furnish not only illusory but even falla- 

 cious results. And, indeed, on a closer inspection of the subject, 

 surmise will quickly give place to certainty. 



* Philosophical Magazine for May 1861., 



