242 Mr Glaisher, On the developments of [Mar. 16, 



(E, I)=l, (2V, 2U) = 2, 



(E, G)=l-k\ (27, 2W) = 2k\ 



(E, 2U) = 2-h\ (2V, I) =1 + h\ 



(E, 2W) = 1, (27, G) =1-F, 



(E, 2V)=l-k 2 , 

 (2U, I) = k\ 



(G, I) = k\ (2W, 2U) = 2-2k\ 

 (G, 2U) = F, (2W, I) =1, 



(G, 2W) = 2k 2 -l. 



Thus, if A x and B x denote any two different series of the 

 seven series 



K x , E x , I x , G x , 2U X , 2V X , 2W X , 



then A X B Z — A 2 B X is always of the form a + bk 2 , where a and b have 

 only the values 0, 1, 2. 



For example, the first equation K X E 2 — KJE X = 1, is equivalent 

 to the identity : 



, 12 12 92 12 o2 R2 - 



{l4 * 2 +^* 2 ^ + 8^ ^ + &cj 



x 1 1 - j* (ad) ¥ - 2^2 (ad) ¥ - #[#[$* (ad) & 6 - &c j 



U 1 „ I 2 - 3 u 1 2 .3 2 .5 | 



-|l-^ /6 "~2 2 .4 2 2 2 .4 2 .6 2 ~ J 



x | ^ (ad) ¥ + ^| (ad) f + i^Jl| (ad) & 6 + &c.} = 1. 



Arithmetical formulas derived from the 21 relations, §§ 42 — 45. 

 § 42. The general form of the 21 relations is 



(a +P X F+P 2 & 4 + P 3 & 6 +P 4 & 8 + &c.) 



x (ft + Qi&tf + <^ 4 + Qb^ 6 + a^ 8 + &c.) 



-OS +Q 1 b 2 +Q 2 k* + Q 3 ^ 6 +<?4 ^ 8 +&c.) 

 x («o +P x p 1 k*+P 2 p. 2 k i +P sPz k e + P 4 p 4 Jc° + &c.) 

 = a + hk 2 , 

 where />„ an<f g w denote the adjuncts of P (l and Q n respectively. 



