of Algebraic Equations of the Fifth Degree. 569 



®2 = -5 (^^la + c I4 + 6 Ii + « I3), 



Elevating each of these eight functions to the fifth power, I 

 now express, as before, 0„ as a function of Py; and !„ as a 

 function of P/z/s,); and observing that (y'„y= 0"4„y, and 

 that [&,j) (^j) = ®'n,/(0.)*, (©"„/) (/3/) = e",.,^(^,), I 

 find &\f + 0'^2,/ + ®"V + @"'i,/ = 



^^6(«®'l,/(/3.0 + ^®'2,/(^..) + ^®"2,/(/3.0 + ^®"l,/(/3 .))' 



1 ■ I- (aa.) 



+ 55(«®"i,/(^.0 + *®"2./(/3.o + ^®'2,/(/3 .) + rf<^'i,/(/3 s))' ; _ 



of which the first member is a rational function of Pyf; and 

 the second is a function, but not a rational one, of Py-/^,\ an- 

 other root of the equation for P. 



In this theorem we may evidently change^ successively into 

 ?, k, I; ff^ e) successively becoming i C^^^\ h T^-^Y / T^^Y 

 We may also \ix\\.ef{^f^ and/' instead of yi But for cer- 

 tain values of the letters in (^}f) (^cd^ . . the equation (aa.) 

 will be discontinuous in consequence of the hypothetical cha- 

 racter of the fundamental theorem (v, w.). 



40. Writing in the expression for 0'^iy + 0'^2 / + 0"^2 f 

 + 0"^/, »?'' e'«,/(/^.) instead of 0'„,/(/j.)', and .?4« 0',,^^^!) 

 instead of &'n,f((it)l and denoting the function which will 



* Compare the first of the equations (y.) with the equations (x.) and 

 (V2.). 

 t For e'^jy + . . is evidently included in the form 



M+ 'v/N+M'+ ^N'+M'- a/N'+M- \^N> 

 or 2(M+M'); 



where M and M' are rational functions of P^. 



