of the Fifth Degree. 51 



take 



and, combining the relations 



1 1 



a = rh, b = 7\^, c— , d= —, 



with the subsidiary equations 



u = r^-] — v=:r„d , 



r rj 



m'= -v/m^ — 4.a, v'= v/v2 + 4d, u^v' = io, 

 the latter of wliich affoi'd 



I find 



2Q, = uv{u + v) + {u-v){2^-w), . . . . (e) 



6^2 = j<j,(j<2^^2)+(2^2_j^9_2^)^^ . . . . (f) 



-l£, = u^-\-v^-Z^ti + v){u^~v^-Z^-w). . . (g) 



45. The symbols r and r^ have been eliminated by means of 



2dr = u + i^, 2dr^ = v + i/, 



and their reduced i-eciprocals. 



46. Since « + w is a value of x, the operation 



5{u + v){{u + v){e) + {i)}+2{g), 



which reproduces the trinomial, suggests that u and v should be 

 separately determined in terms of ■& and x then formed by adding 

 these expressions. 



47. Fixing x and varying d, we find x in terms of different 

 values of ■& which are thus connected with each other. Each 

 particular value of x is, indeed, linked with three particular fac- 



the Petersburg Novi Commentarii. The former may be simplified ; for if 

 a;5— 5P«3_5Qa,2_ 51^ + E=:0, 



x^=ai-\-bv'+cP-'l-di'', and abcd=: — ^, 

 then 



V=ad+bc, 



Q=a^c+ab^+bd^+c-d, 



R=a?b+b^d+ac^+cd^—?^-3y, 



-E=2a^-5PQ- m^ (£^ + ^% £^ + ^). 

 \ c d b a I 



In the trinomial under consideration, I find 



2.y,-2 2.%r2 



\ rir'-i' 



E2 



