135 



ultimately on an equation of the sixth order, viz. (/<o) 5 , (/w 2 ) 5 , 

 (/a; 3 ) 5 , (/w 4 ) 5 are the roots of an equation of the fourth order, each 

 coefficient whereof is determined by an equation of the sixth order ; 

 and moreover the other coefficients can be all of them rationally ex- 

 pressed in terms of any one coefficient assumed to be known ; the 

 solution thus depends on a single equation of the sixth order. In 

 particular the last coefficient, or (/w ./u/ 2 ./o> 3 ./w 4 ) 5 , is determined 

 by an equation of the sixth order ; and not only so, but its fifth root, 

 or /w ./w 2 ./o> 3 ./w 4 (which is a rational function of the roots, and 

 is the function called by Mr. Cockle the Resolvent Product), is also 

 determined by an equation of the sixth order : this equation may be 

 called the Resolvent- Product Equation. But the recent researches 

 of Mr. Cockle and Mr. Harley* show that the solution of an equa- 

 tion of the fifth order may be made to depend on an equation of the 

 sixth order, originating indeed in, and closely connected with, the 

 resolvent-product equation, but of a far more simple form : this is 

 the auxiliary equation referred to in the title of the present memoir. 

 The connexion of the two equations, and the considerations which 

 led to the new one, are pointed out in the memoir ; but I will here 

 state synthetically the construction of the auxiliary equation. 

 Representing for shortness the roots x v x v a? 8 , x# x 5 , of the given 

 quintic equation by 1, 2, 3, 4, 5, and putting moreover 



12345 = 12 + 23 + 34 + 45 + 51, &c. 



(where on the right-hand side 12, 23, &c. stand for x^x^ # 2 ,r 3 , &c.), 

 then the auxiliary equation, say 



(*I* ')=<> 



has for its roots 



^ = 12345-24135, 4 =21435 13245, 

 <p 2 = 13425-32145, < 5 =31245- 14325, 

 3 = 14235-43125, ^=41325-12435, 



and, it follows therefrom, is of the form 

 (1, 0,C,0,E,F, 



* Cockle, " Researches in the Higher Algebra," Manchester Memoirs, t. xv. pp. 

 131-142 (1858). 



Harley, "On the Method of Symmetric Products, and its Application to the Finite 

 Algebraic Solution of Equations," Manchester Memoirs, t. xv.pp. 172-219 (1859). 



Harley, "On the Theory of Quintics," Quart. Math. Journal, t. iii. pp. 343-359 

 (1859). 



L2 



