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XXXVII. On Symbolical Decomposition ; and on the last two 

 Papers of Mr. Jerrard. By James Cockle, M.A., F.R.A.S., 

 F.C.P.S. tyc* 



1. f I ^HE identical equality 



dx dx 

 may be written thus, 



dy 1 dy 



or, symbolically, 



consequently 

 is a solution of 



and therefore of 



dx y dx * " " ' * 



(±-l. d JL\ y=0 - 



\dx y dx}* ' 



y 



y — a 



/ d 1 dot\ 



\dx a. dx)* 1 " ' 



\dx J\dx u dxr * 

 the symbolical development of which is 



-T— -(B+->— \ — 4-^- — 4- l -(— Y-l.— "V -OH) 

 L^ 2 V « dx) dx a dx o?\dx) a dx 2 J 



And if (1) be identical with 



I— 



\dx 2 



{ - +2 4« +r }=°> < 2 > 



we have 



-e*+*l-£ ....... (3) 



a cfo? o?\dx) a dx 2 ' * 



Now a, a particular integral of (2), being known, ft is known 

 by means of (3), and the symbolical decomposition of (2) is 

 effected. On my communication to Mr. Harley of the foregoing 

 solution of the problem of decomposition, with a suggestion of a 

 possible difficulty, that mathematician, with his usual prompti- 

 tude, pointed out that (3) and (4) are not independent conditions, 



* Communicated by the Author. 



