474 DR G. PLARR ON THE DETERMINATION OF 



then the equation xfj'Vidp = decomposes itself into the three 



= Ejdy +Qda, 



= Qdy + B, 2 dz. 



The elimination of j 1 gives the equations 



PjQ-P^^O, 



PgQ-P^O, 



and their consequence 



KiEa-Q^O. 



It is evident that each of these equations contains the three radicals u, v, w 

 all three at the same time, and their transformation into rational expressions 

 depending on the nine coefficients 



Sai , So/ , $>ak , 

 S£t, Sfij, &c, 

 iyyi, &c, 



will lead to hopeless complications from a practical point of view. Of course 

 the expressions of Q, R 1? R 2 may be slightly simplified by the use of the direct 

 function <p instead of </> _1 , but the fact remains that powers of uneven degree 

 of u, v, iv will affect together always each of the equations. 



We will only add the remark that theoretically the two scalar equations 

 above, and the two scalar equations to be derived from 



r it 



will constitute four equations from which the three elements of the versor p 

 may be supposed to be eliminated, so that the problem is theoretically definite. 

 We may also remark that when ^ r (Yidp) = is satisfied, that is when the 

 point of tangence is situated on the limiting curve, then we have for the primi- 

 tive function 



y}r(Yidp)= -dpi^ + Uj . 



As Y(idp) is directed normally to the curve, the equation shows that a rotation 

 c parallel to the normal produces a displacement of the extremity of p in the 

 direction of the tangent. 



From the above equation we draw by the application of the operator i/» : 



f 2 (Vidp)= -fdpCR^ + RJ . 



Now as \p'i = Q we have also ^(k) = 0, where 



K=V(ylr'jyfr'k) generally. 



