Mr. Cockle on the Theory oj Equations. 133 



so that, on developing, we shall have on writing x' . X" for 

 2 (#*') . X (#*"), &c., 



= {t-2s) x {*' . X 1V . (X" + X"') 4-X" . X'" . (x' + X lv ) 

 -A f .X'".(x'' + X l ')-X".^.(x' + A'")} 



+ 2f{(x' + X / ").(A" + A lv )-(A' + ^ v ).(A" + ^)}. . (1.) 

 Let / = 2 w, and s = « — 1, then, if n < 3, the last equation is 

 identically true, but not in any other case. The method of 

 the two first paragraphs, consequently, detects every case of 

 failure; the last-mentioned instance of which is connected with 

 the fact that, implicitly at least, every expression of the form 

 (a.) contains in its right-hand side a term free from x which, 

 with the above values of t and s, vanishes from 2 Y. These 

 values are those which occur in exterminating the 2nd, 3rd, 

 and rth terms of an equation. 



4. If, in the case of w = 2, t-=4t, and s=l, we reject in (g.) 

 the solution ^ = 0, we are conducted to 



(^'-4') 2 (*f-O = 0, .... K) 



having multiplied by the coefficient of A' 2 before commencing 

 our operations. This agrees with what we have inferred from 



5. It seems to follow from this, that biquadratics can be 

 reduced to a binomial, and equations of the fifth degree to a 

 trinomial form, by an expression for y consisting of four 

 terms, determinable by one linear*, one quadratic, and one 

 cubic equation. 



6. At p. 384 of the 26th vol. of this work, I have only al- 

 luded to the equation (3.), which, for cubics, conducts to the 

 reducing equation 



? + **(£) + ££=<>; .... (*J 



\f , / ft ft 



and to a similar one for biquadratics ; but if we discuss the 



equation <p {(A^ + M^ 4 )- 1 } m 0, (3.)" 



it will be found that, though in appearance more complicated, 

 it is in reality simpler than the former, inasmuch as the case 

 of x=0 is not excluded; and if X = and ju. = 1, we have the 

 form actually taken by the reducing equation in my solution 

 of a perfect cubic at p, 248 of vol. ii. of the Cambridge Mathe- 

 matical Journal. 

 Devereux Court, Temple Bar, James Cockle, Jun. 



December 29, 1845. 



* The ' base' equations are linear, as will be seen on referring to my de- 

 finition at note f of p. 126 of this (27th) vol. If the roots of the trinomial 

 equation of the fifth degree are given by the expression b $ (a), then •*/- is 

 contained in the solutions of the functional equation ^ 2 (,a) — az=z 0. 



