r 



TouxG — Algebraical Equations of the Third Degree, 8fc. 

 the first member of which, equation is a complete cube. Hence, 



Aj A3 Az A3 



A, ^ A3 A, '^ \ZA^A3 A^y 

 or, 



X = _ — + 3 Z "^^" ~ ^^0^1 ro] 



2. Suppose now that it is the second of the conditions [1] that is 

 satisfied; then the first condition also will be satisfied, provided a 

 quantity^ be so determined, as that, when it is added to A3, the con- 

 dition 



3(^3+:?^) ^1 = ^2' 



may be satisfied ; that is to say, provided we make 



since this value of p, when added to the coefficient of sc^, will convert 

 the proposed equation into 



— -^- x^ + Aox"' + A^x + Aq =•• px"^. 



ZA 

 Hence, multiplying by —J—, , and observing from the stipulated con- 



-" A ' 



dition [1] that Aq is the same as 7^ , we have 



0A2 



. the first member of which equation is a complete cube. Consequently, 



A, 



^'a)''a?'''- 



.x + ^ =^y3 — i,: 

 -0.3 -CL2 



M^-V3^\p) 



A. • ( ^[ ~ A^ j 



or, 



