and on Co-resolvents. 195 



where 



t'^y^=Y§) ------ -(w) 



and /* is a new functional symbol introduced for convenience. 

 VIII. Now 



du du dy , .dib dy , .du 



'drx^~d^j'i:^~^^'^^T^j'~dz~^^y^~d^. ' ■ ^^^ 



d^ u d { , ^du) d ( ,^du] d { f , ^^^ du 



dx^ 



d^u 



d x^ 



and so on. For we know that 



d f , .dib'\ d r , .du] d ( r . .^^du] . . 

 d d (f , .^^du] d' (f ,.\Hlu] ,, 



d(yd^]d(ydv) 



dx I dz } dz { dx ) 



where v is any function of the independent quantities x and 

 0, and V any. function of v. 



IX. Next, denoting by a suffixed zero the value which a 

 function takes when x vanishes (for instance, denoting by 

 Kq the value of K when a; = 0) we have, by Maclaurin's 

 theorem, 



. cdii^ x , rd^u\ ^^ , D / X 



^ =-^» + I d^J T + [-dUF) „ • O + *"• - (^^) 



Hence, b}^ what has preceded, 



, . duQ X . d f r . A^ duQ} x^ , 



a series whose general term is 



X. "We have now to express Uq or f {y^) as a function of 

 z. For this purpose making x=--0 in (o) we deduce by 

 means of (m), (n) and (o) 



Uo=f{yo)=ff('^o)=ftlcf^^F{yo) + hz7r(y^) -{- x(2/o)) (ac) 



whence 



t~' (2/0) — X (2/0) == ^ (a i^ (2/0) + ^ ^ (2/o)j (ad) 



o 2 



