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XXIII. On a Proposition relating to the Theory of Equa- 

 tions. By James Cockle, M.A., of Trinity College, Cam- 

 bridge ; of the Middle Temple, Special Pleader*. 



J. 1 ET x be the root of the general equation of the wth 

 -" degree, and 



y = A'x x ' + A"x x " + A'"x x '" + A™ *>■";. . . (a.) 



also let m Y be composed of symmetric functions of, and be 

 homogeneous and of the ?«th degree with respect toy; then, 

 if n > 2, 2 Y may be reduced to the form 



(a' 1 A' + fl" 1 A" + S') 2 + (a" 2 A"+i") 2 , . . . (b.) 

 where b' and b" are not both zero. 



2. For, let 



A'".rf + A" a£ y = I' a* + F x*', . . . (c.) 

 then if y r = (A! +'*)«*' + (A" +V)*f + l r , . . (d.) 



1» = (e.) 



Now 2 Y is to be reduced, by means of (d.), to the form (b.), 

 independently of A, or, what is the same thing, of A-f Z; butt 



t Y- S (b4+ft-.U 1 }\ W 



] m denoting a homogeneous function of the enclosed quan- 

 tities of the mtb. degree. And, if n — 1 > 1, 



P l ;.,]«_,]« »0 (g.) 



may be satisfied without making the l's zero. 



3. Following a notation similar to that used in my last 

 paper t, let (p, q) represent the coefficient of A^A^ in the 

 development of 



tp 2 -sp* = zY = 0, (h.) 



p 2 and p x being respectively the coefficients of the third and 

 second terms of the transformed equation in y ; then, if (h.) 

 be reducible to the form (b.), we have 



... + ... +#±*/ZT. b" = 0; . . . (i.) 



and both the values of the above expression can only vanish 



when b' = = b". Substitute for b' and b", equate each ex- 



A'" 

 pression to zero, and eliminate r— between the two ; then we 



have (1, 3) (2, 4) -(1,4) (2, 3) = 0, . . . . (j.) 



where, for instance, 



(l,3) = ^(«')-25SK').S(<"); . . (k.) 



* Communicated by the Author. 



•f- For the process, see par. 3 of the place which I have before cited, at 

 the first line of p. 126 of vol. xxvii. of the Phil. Mag. S. 3. 

 J Phil. Mag. S. 3. vol. xxvii. p. 292. 



