234 



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



The additional term arises from the fact that it is not E and 

 E', I and I', U and U', and V and V that correspond to each 

 other, but E and /', / and E', U and V, and V and U'. The 

 quantities G and G' correspond to each other, as also do W 

 and W. 



Similar formulas in which K and K' are not involved. 

 §§ 33, 34. 



§ 33. The following fifteen equations in which K and K' are 

 not involved are also forms of the relation (i) 



xviii) ETJ' - rv = ik' 2 7r, 



xix) E'U - IV = PV, 



xx) ev - ru = i(i + k ,2 )7r, 



xxi) E'V - IW =i(1 + F)tt. 



The first three equations remain unaltered when k 2 is changed 

 into k"\ but the other twelve equations form six pairs, the equations 

 in each pair being convertible into each other by this change. 



These fifteen equations may be readily verified by expressing 

 all the quantities in terms ofiTand K' and any other two quantities 

 which correspond to each other. Taking for example E and T as 

 the two corresponding quantities all the equations reduce to (i) of 

 § 31, viz. to 



EK' + TK=\tt, 



