THE CURVE ON ONE OF THE COORDINATE PLANES. 469 



where p( )q is to express the operator of a conical rotation, in which pq=l, and 

 Tp = Tq = l. 



For any position of the point of tangence at the extremity of p (generally 

 comprised within the inside of the limiting curve), we represent the variations 

 of the quantities by the characteristic S, reserving the sign d of the differential 

 to the increase of p when the extremity of this vector moves on the limiting- 

 curve. 



In this latter case we assume (having Spi = 0) 



p=jy + lcz; 

 consequently 



dp =jdy + kdz 



represents the element of the limiting curve. 



In the general case p changes into p + Sp ; putting 



e — 28p . x q 



we get 



Ba = Yea, S/3 = Vej3, By = Yey 



Applying these to S(<£ -1 w) we get 



B(<p~ 1 co) = '2Yea. a 2 Saco + XaaPScoYea 



namely 



8(0 - ia)) = V(e0 - V) + " i^Tae) 



This written for o> = i, j, k, gives first 



2uSw = Si8((f>- 1 i) 



= St[Ve0- i ; + 0- 1 Vie] 

 = -2S.eYi<t>-H 



Hence (and mutatis mutandis) : 



Bu = S . eid>~H 



u 



Bw=-—Sekd>- 1 k. 



With these expressions we get 



Be=-t-Seicf>-H, 



u 



8 Pl =±(Ys4>-H + 4>-iYie)+^S.<d<f>- H, 

 and the second members evidently represent each a linear vector function of €. 



