Sir William Rowan Hamilton on Equations of the Fifth Degree. 



375 



v.,. = — ■ 



365 



(174) 



The three first values of v may therefore be thus collected : 



TfTV345=-3; tI7V453=9-4d; ^|^v334 = - 6 + 4d ; (175) 



and the three last values, in an inverted order, may in like manner be expressed 

 by the equations : 



^V435 = -3; ^|^v^3=:9 + 4d; ^^y^=-Q-\j,. (176) 



36. It Is evident that these six values of v are of the forms (113) and (114), 

 and that they verify, in the present case, the general relation (121). They shov? 

 also, by (c) and (d) of article 28., that not only h^, but h„ vanishes in this ex- 

 ample ; the common value of the two sums (121), of the three first and three 

 last values of v, being zero. Accordingly, if we compare the particular equa- 

 tion (147) with the general forms (1) and (2), we find the following values of 

 the coefficients (b, c, d, e, not having here their recent meanings) : 



A = 0, B = _ 5, c = 0, D =: 4, E = 0, (177) 



and 



p = — 5, y = 0, r = 4, 5 = 0; (178) 



and therefore the formula (51) gives here 



H, = 0. (179) 



We find also, with the same meanings of Q and f as in former articles : 



tIt (V345 + 0^v,,3 + 0v,3,) = 3 (40^ - 0) + 4f ; 



29« 

 126 



(V354 + e'va43 -f 0v,3,) = 3 (40 _ e^) -f 4f ; 



and, therefore, by (c) and (d), 



2^ 2? 5-' (H3 + x/hJ = {3 (W -6) + 4^r, 

 2' 3^ 5-« (H3 - ^/HJ = {3 (46 - 6'-) + 4^Y ; 



equations which give, by (11) and (57) : 



/H,= 2-=5'»(0-0^)(23-f3f); 



and 



H, = -2-^3' 5^»( 197 + 69d- 



(180) 



(181) 



(182) 

 (183) 



