252 Sir William R. Hamilton on the Argument of Abel, 



two new tables of double entry, which the Avriter of the present paper has had the 

 curiosity to construct and to apply. 



[27.] In general, if we change x io x-\- ■/ — 1 j/, and Oj to a,-}- y'— 1 b^, 

 the equation of the fifth degree becomes 



(a? + V~i yy + (a, + /=i h) {x + V~\yr 



and resolves itself into the two following : 



I. . . ^— 10^V + 5a:y 



+ a, (a;*— 6 x^y^ +y) — i, (4 x'^ — 4 a;^') 



+ o,(a;^-3a;y)-5,(3^j/-y) 



+ a3(ar— /) -2^30;^ 



^^a^x — b^y^¥a^z=.0', 

 and 



11... 5^V-10ar>'+y 



+ a, (4 ar'^ — 4 xy^) + 5, (a:^ — Qx^'y^ +y) 



+ a^ (3 x'y - y) + 6, (a;' - 3a;y) 



+ 2a^xy + b^{x''-y'') 



+ 04?/ + 6^0:4-65 = 0; 



in which all the quantities are real : and the problem of resolving the general 

 equation with imaginary coefficients is really equivalent to the problem of 

 resolving this last system ; that is, to the problem of deducing, from it, two real 

 functions (x and^) o/'ten arbitrary real quantities a^, . . .a^,b^, . . .h^. Mr. 

 Jerrard has therefore accomplished a very remarkable simplification of this 

 general problem, since he has reduced it to the problem of discovering two real 

 functions of two arbitrary real quantities, by showing that, without any real 

 loss of generality, it is permitted to suppose 



a, = Oj =: O3 = ftj = 6j =: 63 = 64 = 0, 

 and 



a, = 1, 



Oj and b^ alone remaining arbitrary : though he has failed (as the argument de- 



