1885.] K', E', J', G' in ascending powers of the modulus. 233 



the six equations 



(i) EK' + I'K = lir, 



(ii) IK' + E'K=-\ir, 



(iii) OK' + G'K = hir, 



(iv) UK' + V'K = hir : 



(v) VK' + UK = lir, 



(vi) WK' + W'K=\tt. 



Each of these equations is thus a form of (1), the three terms 

 being reduced to two. The last three equations are immediately 

 deducible from the first three by addition, since 



U=i(I+Q), V=i(G + E), W=±(E+I). 



Equations (iii) and (vi), which involve G and W, are perfectly 

 symmetrical. The other four formulae exhibit the correspondence 

 between E and /, and between U and V, which is observable in so 

 many other results. 



Dividing by KK', the six equations become 



EI' 



(») If + 



K ' K' 2KK" 



IE' IT 



(iv) w + 



K K' 2KK" 



*L %- _ 77 L_ 



K + K' 2KK' ' 



U V TT 



iv A" 2KK" 



> v ) w + 



K ' K' 2A7T 



(vi) ■= + 



K ' K' 2KK' * 



§ 32. If we express the relation (1) in terms of K and one 

 other letter, the expression equated to \tt contains three terms 

 except when the second letter is G or W. 



The group of four formulas to which (1) belongs is 



(1) EK' + E'K - KK' = \ir, 



(2) IK' + I'K+ KK = Itt, 



(3) UK ' + UK + IKK' = Itt, 



(4) VK ' + V'K - IKK' = \ir. 



