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XXXIV. On the Existence of Finite Algebraic Solutions of the 

 general Equations of the Fifth, Sixth, and Higher Degrees*. 

 By James Cockle, M.A., Cantab. ; Special Pleader-f. 



7. \\T HEN y - A' x x ' + A" x x " + . . + A xi .r xXi . . (n.) 



* V and $ r m*M-.*Al * & 



h v and A 2 having the forms of the quantities squared in (b.) J, 

 what is the limit of n? 



8. Make 3 Y M = /* 1 3 +j'A' + 3L" (p.) 



then§ j' = J? } + Ji 2) + &c; (q.) 



but Ja 2) = and A", A'", disappear from j', if] J 



y = A!x x ' + L" + L iv + L v + ... + L» . . (r.) 



and I/»=A» (**"-*=. **''') (s.) 



So, £'' = /* 2 3 +y'A'' + l- (t.) 



9. Let y r = Lf 8* + Z/» (*f - & *f ) + I, . . (u.) 

 1„ = 0, L = A -f /, and Z = a constant, then^f, 



o=y = [i 1 .u_ 1 ]i 2 t 



o=j"=[l 1 .l w _ 1 ] 2 2 k («.) 



= liv = [l 1 .l ra _ 1 ]3 J 



.*. « — 1 > 3, or 7i > 4-. . . . . . (v.) 



10. Again, j' — is equivalent to** 



y* A iv + . -f- y xi A xi = 0-j 

 y*A"+. + y xi A xi =0 I 



yfiA™ + . + y xi A xi = [' ' K ' 



7fA x + yfA xi = 0j 



j" = 0, on eliminating A 2r+1 , toft 



-y^+.+^A-^Ol 



via y 9 A a + x y 2 A x = Oj ' ' * ' K7 '' 



l iv = 0, on eliminating A vi , A x , to J J 



wyi A iv + viU y A viii = (8.) 



* See my presumed solution of the equation of the fifth degree, at page 

 125 of the last volume of this Magazine. I there used the ratios z u z 2 . . . 

 of the quantities A', A", .. to one of their number, but have here employed 

 other ratios, or, more properly speaking, the quantities themselves. — J. C. 



f Communicated by the Author. J Phil. Mag., this vol., p. 132. 



§ Ibid. S. 3. vol. xxvii.p. 126. j| Ibid. p. 293 (16.). 



IT Ibid. p. 126, note J, and this vol., p. 132, par. 2. 



** Phil. Mag. S. 3. p. 126, line 9. ff Ibid. (/.) and (g.) 



XX Ibid, ifi.) 



