230 Dr Baker, On the differential equations 



rw 

 and putting I </> (w) dw=-\jr (w) + const., 



this gives 



where w 3 , m 2 stand for + u x + const., + u 2 + const., and hence 

 x = ±X 4 cosech 2 [^ J\ 4 (u 2 + yjr (i*i))], 

 y = {\<j> (%) cosech 2 [| Jx 4 (u 2 + -\/r O x ))], 



from which # follows without ambiguity by means of 

 x (xz - y 2 ) + {X 3 xy - lX 4 y 2 



= - \x> Jx 4 coth [1 Jx 4 (u. 2 + ^ (ui))] *W . 



From these the functions £ 2 (z^, w 2 ), £i(/M 1; u 2 ) given by 

 — d% 2 = #cfoi 2 + 2/cZmu — cZ£i = 2/c?m 2 + zduq. 

 are at once found to be, save for additive constants, 

 £ 2 = 1 Vx 4 coth [1 V^ (w 2 + yfr ( Ul ))], 

 Si - i v\ coth [| V\ 4 (m 2 + fW)] <£ (Wi) + i\ 3 '»KO -i^ 4 % («i) 



rw 



where % (w) = I (<f>(w)Y dw + const. 



TAew i/ie integral function 



a (u 1} u 2 ) = exp ( I % 2 du 2 + Zidua] 

 is seen to be 



, N _ sinh [I \fx 4 (^ 2 + ^(uj))] -[^^[^^[-i^^i+ix^CM^j+^^+^Mj-i-c 

 |VX 4 



where A, B, G are arbitrary constants, and u^, u 2 are used for 

 ±u x + const., ±u 2 + const. 



We take particular cases of this. 



en X 4 is not zero and th( 

 1 is, save for a factor of t] 



o- (u lt 11.2) = eiV^ jj ( u j + e -W*4«i K (ui), 



(i) When X 4 is not zero and the other coefficients are general 

 the function is, save for a factor of the form e Aui+Bu2+0 , 



