Mr. J. Cockle on the Congelations of Analysis. 533 

 and let the result, divided by u, be 



then we have, by the aid of Leibnitz's theorem, 



(Ml£, l)«=/3«, (5) 



(1, Z>, c£ — , iyu=<yu } . = .... (6) 

 ax 



' ■ ' (1,£, C, d£y, l) 3 u=$u, (7) 



ax 



and so on. Developing (5), (6), (7), we have 



u ! + bu=/3u, (8) 



u n + 2bu! + cu=yu, (9) 



u ,n -{-3bu" + cu'-\-du=;8u (10) 



Differentiate (8), and by means of the result, combined with 

 (8) and (9), eliminate u" and iL We find, on rejecting the 

 factor u } 



'p + V-c^+P-c (11) 



Again, differentiate (8) twice and subtract the result from (10). 

 In the difference substitute for u" its value obtained from (9), 

 and, by means of (11), eliminate j3'. We find, neglecting the 



2b*-3bc + d-b'i = 2l3 3 -3t3y + $-/3". . . (12) 

 Hence the critical differential functions 



2b 3 -Sbc + d-b", 



remain unaltered by the above transformation from y to v. 

 In the algebraic equation 



(1, b, c, d, ..Jy, l) n =0, .... (a) 



which yields (a) when exponents arc changed into differential 

 coefficients, let v-\-u be substituted for y } and let the result be 



(1, «, ft 7i * • -X v > 1 X=0; . . . . (b) 

 then 



(1,SX«»1)=A (13) 



(1, b, cju, l) 2 =y, (14) 



(l,b,c,<r£u,lf=8, (15) 



