240 Mr Olaisher, On the developments of [Mar. 16, 



and taking any letter, say for example G, the equations give the 

 values of 



(OK), (GE), (GI), (GU), (GV), (GW). 



Definition of the Adjunct, § 39. 



§ 39. The series for K r , E 1 ,...,2V 1 ,2 W l and K 2 , E 2 ,...,2 V 2 ,2W 2 

 in ascending powers of & 2 were given on p. 192 (§ 9). 



The coefficient of k 2n in any series A 2 is connected with the 

 corresponding term in the series A x by the curious law explained 

 in § 4 (p. 188). Thus, taking for example the letter E, the coeffi- 

 cient of k e in E x is 



r . 3 2 . 5 



2\4 2 .6 2 ' 

 and the coefficient of k G in E„ is 



_1 2 .3 2 .5 /2 2 2 2,1 2 



~ 2 2 . 4 2 . 6 2 VI ~ 2 + 3 ~ 4 + 5 ~ 6. 



The quantity in brackets may be conveniently termed the 



l 2 . 3 2 . 5 



adjunct to the coefficient ' ' and the above expression may 



be written 



1'.3'.5 

 2 2 .4 2 .6 2 W ' 



Tfo series for K x> E t , &c, and K 2 , E 2 , <tc, § 40. 



§ 40. The series for K 2 , E 2 ,...2V 2 , 2W 2 are derivable from 

 those^ for K t , E 1 ,...2V 1 , 2.W l by appending to each term after the 

 first its adjunct. The first or constant term is either 0, 1 or 2 

 and follows no regular law. The coefficients of Ic 2 in 2 U t and 2 U 

 are anomalous, being and — J. With these exceptions, all the 

 series with suffix 2 differ from the corresponding series with 

 suffix 1 only by the addition of the adjunct to each coefficient. 



Using the notation explained in the last section the fourteen 

 series given on p. 192 may be written 



12 12 02 -12 OS £2 



K < = 1 + ¥ F +2ri- k '+ ?4t«. *■+*«, 



l 2 l 2 3 2 l 2 3 2 & 



k = 2* (ad > 7 " + kl 2 < ac] ) ** + k^f- m 7 " + &c -> 



