162 THE ROYAL SOCIETY OF CANADA 



(19) H^^^ =0, K^^^ =A<t>\ 



The first and minus first Laplacian transformations of f and f ' are* 



ôlog , — I — 



dv ou 



a log 



dv du 



The relations (17) shew that the pencils of double directrices P(ô') 

 and P{h") are transformable into each other by Laplacian transforma- 

 tions. 



We shall now proceed to the integration of the Normal System. 

 It follows from (17) that 



(20) -^ (-^\ = -^ = ^ (X=.\ ■ 



dv \ J (f, J J fj) du \ J ^ J ' 



and that therefore the expression ( -f-= — j is a complete dif- 

 ferential. 



Since H^^^ and K^^^ vanish the second equation of (18-a) can be 

 written 



du\ dv ^ ^ dv y""^ J <i> J) ^ d / 1 \ 



and the second equation of (18-&) can be written 



dv \ du du \ sT^/) _ _d_ / . 1 \ 



On integration these equations give 



(21) dv ^^ dv V^ JJ-J~ J~T' 



du du\ J <!> J J <i> ' 



*N, p. 487. 



