of the Fifth Degree. 361 



where ^' and ^" are fimctions of the same form. Similar trans- 

 formations exist for t^, &c. 



24. Restoring the values of b, c and d, we find that the roots of 



are i, ? and i' (excluding i^), and consequently that f takes the 

 three forms 



^-i^\, K'^-^'^\ KU-n\, 

 corresponding to the roots i, i^ and i^ respectively. On making 

 the proper substitutions for f we find 



^'3 = 2/« + 2/y - (»■ + i^)y^> ^"3 =yp+ys- (* + «^)yy> 

 ^^=y«-{i+i')y,+i'yy, ?"4=y^-(«'+'')yv+»V 



25. It will repay us to rest for a moment in order to ascertain 

 the effect of supposing ^' and ^" to vanish separately. We shall 

 at once verify our results and illustrate their nature. 



(a) Let ^1 and ^\ be zero. We find, on expressing y in terms 

 of /, u and v, 



^"j = (2j4-z-i> = 0, 

 equations which can only be satisfied by supposing u to vanish. 

 In this case therefore the quintic in y is of the solvable form of 

 Euler. 



{b) Let ^'3 and f"^ vanish. We are led to 

 ^'^ = {l-i-i3 + i*)v = 0, 

 ^>^={?-t^-l + i)v=0, 

 and, consequently, to the solvable form of De Moivre. 

 (c) Let ^'4 and ^"4 vanish. Then 



?'4 = (l-i2_i4 + i)^=0, 



and we are reconducted to the quintic of Euler. 



{d) On eliminating y^ and ys from 2 . j/^ by means of Bi=0 

 and ?=0 respectively, we find 



^•^ - 1+i ' 

 and from (a) and (c) we obtain 



^,^4=?",?"4=5(l+i)^U. 



Hence, combining these relations and reducing, 



This property, in the discussion of Euler's quintic, enables us 

 immediately to depress its final equation to the twelfth degree. 



