Vector Differentials. 189 



in which if we interchange p and p\ q and q\ r and r f , we 

 shall change yjr into yfr'. 



To build up this function notice first that the quantity c 

 is the absolute curvature of the orthogonal trajectories of the 

 given surfaces. If c x be the tortuosity of these curves then 



1=^ d> 



cf. Tait's ' Quaternions/ §§ 299, 300. Hence 



die ix) 

 tin 



do d ub 



= — At + c j 

 on an 



= ^ x + £^ W 



which gives definite values for q and p 1 , and shows that R 

 vanishes. Again 



dc = dTY\Jv 



= S^dp, by Tait, § U0 (1) 



= — Sdpyjr'fMj 

 so that 



cfc t^c (/<? 



<w dm dn ' w 



giving values for Q and r. Next take 



^dp = dY\Jy 

 = d(y X v) 



ss — Yyv^dp -f Yvcpdp, say ; 



then by taking conjugates, 



yjr'dp = ^' Vx y dp — <j>'Vvdp 9 

 whence by putting v for c/p and remembering that x^ — x'l-^ 



= cXSA%M + C/,6S/Lt^At, ^-1) 



giving values for ^ and q'. 



