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



Expressions for the quantities E, I, G, U, V, W in terms of 

 any one of them, § 46. 



§ 46. In § 8 (p. 191) the six quantities E, I, G, U, V, W were 

 expressed in terms of E and of /. It seems worth while to give 

 the expressions of these quantities also in terms of G, U, V, W. 

 The complete system of formulae is as follows : 



E=E =I+K, 



I =E-K =1, 



G=E-k' 2 K = I+k 2 K, 



U = E- 1 (1 + Jc' 2 ) K=I + \h 2 K, 

 V = E- i k' 2 K =7+1(1 + k*)K, 



W=E-\K =I+\K, 



E = U+±(l + k ,2 )K=V+±k' 2 K, 



W = G + l{k' 2 -k 2 )K=W. 



Change of q into — q, <f and q*, § 47. 



§ 47. The following table shows the transformations into each 

 other of the quantities k, k ', K, K\ E, E', &c. produced by the 

 change of q into — q, q 2 or <f. 



