394 Professor Sylvester on the Solution of 



{(4B 2 -D) 2 -26(4B 2 -D) + ^ 2 



-2{2(4B 2 -D)(BD)-c(4B 2 -D)-26BD + 4^ 

 + 4B 2 D 2 -2cBD+/=0. 

 Hence we may write 



(4B 2 -D) 2 -26(4B 2 -D) + d=\, 

 2(4B 2 -D)BD-<4B 2 -D)-2£BD + e=\B, 



4B 2 D 2 -2cBD+/=^ D ; 



from whicli equations B and D are to be determined. Elimi- 

 nating X between the first and second and between the first 

 and third of these equations, we obtain two equations, of which 

 the arguments are 



D 3 ; B 2 D 2 , D 2 ; B 4 D, B 2 D, BD, D ; 1 



for the one, 



BD 2 ; B 3 D, BD, D; B 5 , B 3 , B 2 , B ; 1 



for the other. 



Eliminating D by the Dialytic method between these two 



equations, we shall have (using points to signify unexpressed 



coefficients) the following three linear equations in D 2 , D, 1, 



viz. : — 



• BD 2 +(.B 3 + &c.)D + (-B 5 + &c.) = 0, 

 . B 3 D 2 + (• B 5 + &c.)D + (• B 7 + &c.) = 0, 

 . B 5 D 2 + (• B 7 + &c.)D + (• B 9 + &c.) = 0. 



Hence in the final equation B rises to the 15th power ; 

 and by combining any two of the above equations, D is given 

 linearly in terms of B ; and, finally, x is known from the 

 equation 



_ :Qt> + D-4B 2 -26)( g + 2BD) 

 x -(4B 2 -D) 2 -2(4B 2 -D)W 

 and has 15 values. 



A like process may be extended to a unilateral equation (of 

 the Jerrardian form) of any degree, say x^ + qx + r^O. 



Introducing the auxiliary equation with scalar coefficients 

 as before, viz. 



# 2 -2B + D = 0, 



x may be expressed as a function of q, r, B, D; and the term 

 containing the highest power of B in the equation for deter- 

 mining B (of which D is a one-valued function), when <w = 4, 





