of Algebraic Equations of the Fifth Degree. 549 



jnerally be € 

 fore have separately 



cannot generally be equal to — r— '=-, , We must there- 



1 + * v^ + «^Vy+ «3vs = 0, 

 1 +'^v^+ .3vy+,2v3 = o. 

 If now we multiply the first of these equations by <^, and 

 from the product subtract the second ; we shall find 



where ^ may have an unlimited number of different values 

 assigned to it. 



Hence, if we cause Vy to disappear by making ^+2 = 3, 

 there will result 



v'^ = vj + a^ 



a, denoting — -2—5; and if ? + 1 = 4, we shall have 



V = «/ "s + 1, 

 3 the coefficient of vj being evidently equal to— a,. 



Finally, on returning to the expression for which 



^« + " 

 involves y^, y^, . . , and making the requisite substitutions in 

 it, we shall obtain 



which, if = 0, will give, independently of vg, 



y^+ y^ + ^.yy^o.j 



And we see that the values of ^Jj and p^ which satisfy this 

 pair of equations must be such as to fulfil the condition <E> = 0, 



o»' l^^y^ + f^i3 2/^ + FyVy + /*jys + l^ty, = 0> a"d consequently 

 to satisfy every one of the ten equations belonging to the same 

 system. 



7. We might now by means of these two equations, which 

 involve p^, p^, x^^ .^•^, Xy^ ^3, and which are both of them of 

 the first degree with respect to py^ and j^^, express pj and j^g ^s 

 rational functions of .r^, a?^, Xy^ xy, and then, from discovering 

 the number of different values which the expression for p„ 

 (either j^i or^g) would assume if the five indices I, 2, 3, 4, 5 



