[28 ] 



VIII. On some Formulcc which seive to i7idicate the limits 

 of the application of Indeterminate Methods to the solution 

 of certain Problems. By James Cockle, Esq., M.A., of 

 Trinity College, Cambridge ,- Barrister-at-Law, of the Mid- 

 dle Temple. Second afid concluding Part*. 

 [The first part will be found at pp. 181-183 of the preceding voliiraet.] 



LET x^, x^, . , Xn denote the roots of 

 a;" + a/x« - ' ^- A'" - 2 + . -I- x'"'> = 0, 

 the general equation of the n\\\ degree, and, by a similar no- 

 tation, let j/„ yo, . yn he the roots, and if, y\ . y") tlie coeffi- 

 cients of the equation in y, when 



y = f (0) + l^x + r'*^ + . + ^^" ^"'> 

 it is required to point out how to determine f'''^ 0', . , . , so 

 that we may have 



y = 0, y' = 0, y" = 0, andy"' = 0. 

 By means ofy = 0, eliminate ^ "' from the three succeeding 

 equations, and, conformably with the notation adopted in one 

 of my previous communications to this work|, represent the 

 respective results by 



2Y'„=0, 3Y'„=0, and4Y'„=0. 

 Next, bv processes which I have already referred to in the 

 present periodical §, let 2Y'„ = be put under the form 



where /^i = x', ^ + ''"i 0" + • + ^^^^ l^\ 



and, in general. 



We have twenty-two quantities (0) at our disposal, — make 

 ^^2 + ^=0, .(A.), V + V = 0, . (B.), 

 and h^ + //g^ = o, . , . , h^^ + hl^ = ; 

 with the aid of the last nine of these equations (previously put 

 under the form 



h,_i± V_l7;^ = 0), 



let nine of the ?'s be eliminated from the first two; after the 

 elimination there will remain 22 — 9 or 13 of the quantities ^', 

 0", . ^xxii still undetermined. 



Now it is known \\ that, whatever be the number of the 

 (|uantities ^, the above expression for y is equivalent to 

 y=S">' + S'x + 3"x^ + . + St"-'^ x"- \ 



• Communicated by Sir George Cayley, Bart. 



t Phil. Mag. S. 3. vol. xxix. J ibid. vol. xxviii. p. 191. 



§ Il.id. vol. xxvii. pp. 126, 292, 29.3. 



11 Sixth Report of the Britisli Association, p. 301 ci seq. 



