1885.] K', E', J', G' in ascending poivers of the modulus. 241 



1 1 2 O 1 2 03 - 



77 — 1 _ _ 7- 2 _ ' 7.4 ' ° ' ° U> ?rn 



^ - 1 2 , « 22 _ 43 '" ~ ' 2 2 . 4\ 6 2 ' 



E 2 = l~\> (ad) Zr - ^| 8 (ad) ¥ - 



1 1.3 



J i = ~ 2 ** ~ 2T¥ ^ ~ 2 2 .4\6 7/ ~ &C -' 



l 2 =l~l (ad)//- £j-| (ad)//- ~p7^ (ad)tf-&c., 



10 12 1 2 .-)2 



®i = 2~ "'" "2^ 4 ^ "*" 9 2 4 2 6 ^ "*" ^ C "' 



^ 2 = 1 + |° (ad) /v 2 + -74 (ad) *■ + 2 f 4 f g (ad) // + &c, 



"I 2 -i 2 02 



2JJ = 2- \¥ - ^~ (ad)* 1 - ^^ (ad)F-&c, 



27, = 1 + ^ 7< 2 + ^ &* + ^i^ *• + &c, 



27 2 = 2+^(ad)/, 2 + ^ (ad)* 4 + ^^ (ad)// + &c., 



9 w _ i _ * • ^ 7.2 _ !"• 3 • 5 7,4 1 2 .3 2 .5. 7 , 6 . 



13 1 2 3t 1 2 ^ 2 ^7 



2 F 2 = 2 - -^(ad) F - ±^- (ad) tf - ^ ^ ^ (ad) /? - &c. 



The 21 relations connecting the 14 series, § 41. 



§ 41. The equations in § 38 form a very remarkable series of 

 relations to which these 14 series are subject, viz. denoting, as 

 in that section, A 1 B 2 — A 2 B 1 by (AB), the 14 series are connected 

 by the 21 equations : 



(K,E) = 1, (K,2V)=2, 



(K, I) =1, (K, 2U)=2, 



{K, G)=l, (K, 2W)=2, 



