310 



SIXTH REPORT — 1836. 



On the other hand, the second part of the process cannot 

 conduct to this useless form for y, unless the first part of the 

 process has led to functions, 4> {x), x (^)3 connected by a rela- 

 tion of the form (36.). This consideration suggests the intro- 

 duction of the following new system of equations of definition. 



N' sy^ + N"*,^'"^ + N'" *„(•"') + Ws^^ + N^ s^^ 



= P\, 



N' .('') + N" s^''^ , + N"' s^'"\ + N'v ,(v'^') + Nv ^(.^) 



— 1)" 



— J^ m-V 



(94.) 



N' L^''^ + N" L^*^'^ + N'" L^'"'^ + N»vl('") + N^ L^'''^ = N, (95.) 



= p; {9^-) 



Pm-l 



P\ - P'P, = 9'o» V\ -P'pl = H\, "• P\-2 -P' Pm- 



OT— 2 



(97.) 



(98.) 



to be combined with the definitions (46), (47), (48), (49), (54), 

 (55), and with the following, which may now be conveniently 

 used instead of the definition (50), 



A + M + N = L 



In this notation we shall have, as before, 



P'o = 9o + PPo^ P'x = 9\+ PP\> '"P'm-2 - qm-'. 

 + PPm-2> P'm-\= PPm-\> 



and shall also have the analogous expressions 



P"o = ?'o + P'Pq^ P\ = Q\ + P'P\ » —fm-'i = ?'»«-2 

 + P'Pm-^^ P"m-\ -P'Pm-\> 



the expression (75) for y will become 



y=f{x)=Po+ p'o + P"o + iPl + P\ + P"l) ^ + 

 + iPr, 



tl At Is, by (56) and (99), 



+ L X :. 



'} (56.) 

 '} (99.) 



+ p'o + p"o + iPi + P\ + P\) ^ + '•'\ noo ) 



