194 Mr. G. B. Jcrrard on some Objections 



particular six-valued functions represented by 



a, a 5 , 



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 



* 5 =(«) 5 . 



4. That there is something exceptional in the case in question, 

 we may see at once without entering into the calculations. 



For beside the equation 



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 suffi- 

 cient for the purpose intended, — we have, in the present instance, 



v=u 5 (E,) 



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

 coefficients of (e x ) ? 



Again, in ordinary cases we can generally obtain a single defi- 

 nite equation of the form 



u = v 6 +v 4 v + v 3 v 2 + . . + v v 5 ', .... (e 2 ) 



but here we set out with supposing that 



u = V / 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, Lagrange's 

 theory should furnish not only illusory but even fallacious results. 

 And, indeed, on a closer inspection of the subject, surmise will 

 quickly give place to certainty. 



5. Combining (e } ), (E x ), we have 



0=fJL 5 + /j, 4 u + fJi 3 u'*+.. + (/j, Q —l)u 5 ; . . (e'J 



in which fju 5 , fju 4 , . . /jl q must all of them, in accordance with what 

 has been stated in art. 2, be expressible as rational functions of 



But as (U), the equation on which u depends, is rationally 

 irreducible below the sixth degree, the coefficients 



/*s> A*4>««(^o— !) 

 of the equation (e\) ought severally to vanish. 



"When, therefore, u } v are equal to a, a 5 respectively, (<?j) ought 

 to become 



v=u 5 , 



which is precisely identical in form with the equation (Ej) itself. 



