564; Mr. G. B. Jerrard's Reflections on the Resolution 



Vk» Vk(12)» • . Vk(15)» y . . . . (o.) 



"^L' ^L(12)» • • ^L(15); 



which, except in particular cases, will be distinct one from 

 another. 



30. But if we designate by Wy /ai^x/cd) . . the function 



(Py.+ P^.)(al))(cd).. , 



which evidently does not admit of more than six different ex- 

 pressions, and observe that 



we shall see that the resolution of the equation V^^ + C, V'"* 

 -f . . = may be reduced to that of a determinate equation 

 of six dimensions, 



W6 + D, W^ + D^W^ 4- . . + De = 0, 

 the roots of which will be 



W„ W,.^ 



31. Could we solve this equation for W, the roots of the 

 general equation of the fifth degree might indeed be easily ob- 

 tained. For from the expression for Py + P/' we might de- 

 duce that for Vf or P. j9j, ^5, p^ would thus become known. 

 And by combining oe^ +p^ ^v^ -^Pci^ +Pz =3/ with x^ + Aj x'^ 

 + Agi^ + . . + A5 = 0, we should be conducted to 



^ = 94 + %!/ + 9^zf+9if + 9o^^'^ ' • • (q-) 

 where g^, q^, . . qQ are rational functions ofp^j^gji^a' ^^ simply 

 of P ; and where 



1/ = pt + p'^Uy 



(p tY and (p"* uY being, as is well known, the roots of the equa- 

 tion 





{t^f+A\{t^)-(^^y=0y 



and consequently admitting of being expressed in terms of A'g 

 and h\, which also are rational functions of j^j, jWg' Pa* 



Section VI. 



32. We have not hitherto taken into consideration the 

 forms of the functions denoted by t and ic. 



