SURFACE MOTION FOR ANY POSITIONAL FIELD OF FORCE. 161 



resulting curves the associate system corresponding to the given initial 

 element. Our first query is: what is the locus of the foci of the 

 osculating parabolas of the ooi curves of the associate system? ' To 

 assist us in answering this, we shall choose the tangent plane as the 

 XOY plane, the point as the origin, and the OZ axis as the normal 

 to the surface. We shall further choose the x- and y- axes as the 

 tangent lines to our isothermal v- and u- parameter curves, respec- 

 tively. 



Here z = 0. The trajectory determined by (o, o, v', v" , v'") is 



associated A\4th the curve determined by I o, o, — , — , — ) . The 



\ dx dx^ da?/ 



latter derivatives may be expressed in terms of the former by using 

 the equations of the surface (1). Thus we have 



^lyj X Xu I" •^\'^ f '^ "^uu ~i -^^'uv^ r" Xtjj)V ~ ~\~ x^v , 



X X-UUU I OXiiUV*^ I OXuVV^ I ^VVV^ \ OX-uyV ~r~ OXyyV V ~j~ XyV f 



with similar expressions for y', y", y'" . Then, 



Ay Vu + yv f' 



(20) 



(X*C ^v, ~1 *t''u V 



By our choice of isothermal parameters, the tangent of the angle 

 between the initial element and the «- parameter curve is i;', so that 



(21) X.0 = 0, yu = 0, a-„ = ?/„. 



Differentiating (20) and using (21), we find 



dx 



(22) 



^ = 1 "+ 

 dx'^ Xu 



where a, h, and c are functions of u, v, v' only. 



Now the coordinates (a, j3) of the focus of the osculating parabola to 



1 / dy dry d?y\ . , 



the associate element I o, o, — , — , — I are given by 



