Mr. Cockle on the Finite Solution of Equations. 191 



1 1. Let* L iv + L vi + . . = ;/ +/, x+ . +p\ n ^ x n ~\ (w.) 

 tkenf, iv S xi .L=(l+^)(// + ..) + . + ?„_ 2 ^- 2 ; . (x.) 

 and we have, in general, n quantities 1 + p, q^ . , q n ~% to sa- 

 tisfy 3 homogeneous equations = j' — j" = l iv , or, n quan- 

 tities p'wP'i) .,2^n-\ lo satisfy 3 other homogeneous condi- 

 tions (y.) and (8.), using the former n quantities to satisfy the 

 group ((3.), non-homogeneous with respect to them; hence 



rc>3. ........ (y.) 



12. (v.) and (y.) are not inconsistent, for, if 



A - l Al _ n ~ r + m /_ i 



^"^A^- nm m 



and 



r Y' n =p r — A 1 ;j r - 1 ?i + A 2 p r _ 2 p 1 2 + . • -^ — %-iK> ( aa -) 



then this accented value of Y is a critical value from which 

 7 ; o» %">'•' disappear, and, since (2.) and (3.):j: are respect- 

 ively 



3 Y' 5 = 0,and 4 Y' 5 --i- 2 Y 5 '2 = 0, . . . (ab.) 



my solution would fail if n — 1 were < 4. Hence, for all cri- 

 tical functions (y.) degenerates into (v.) ; and, after solving 

 critical equations, we shall have quantities ^/ , q . . left for sa- 

 tisfying (other) conditions whose degrees are unaffected by 

 our previous operations. 



13. However numerous might be the groups (/3.), (y,), 

 (8.), or the relations forming those groups, it would seem that 

 some of the A's being lost at each descending step, the limit 

 will not be proportionally elevated. We may make 



2 r = (l+i>)('-%- 1 + .. + ?Ll 2 * n - 2 , . . (ac.) 



for the A's introduced as we ascend the groups. 



14. A' and A" will introduce new values of 1 +p and q into 

 the (now to be combined) equations (2.) and (3.), or (ab.), but, 

 as it appears to me, no new difficulty. The y's may be de- 

 rived from the t's§, and from each other by one operation 0, 

 and if 



y = © ( T ), then y = (y), &c. ... (ad.) 



Grecian Chambers, Devereux Court, James Cockle, Jun. 



January 31, 1846. 



* See Sir W. R. Hamilton's " Inquiry, &c." into Mr. Jerrard's method 

 (Sixth Report of the British Association) from [4.] p. 301 to line 9 of 

 p. 304. 



f Ibid. p. 303. | Ph'l. Mag. S. 3. vol. xxvii. p. 125. 



§ Ibid. pp. 292, S>93. 



