EussELL — Geometry of Surfaces derwedfrom Cuhics. 463 



Let f, r], {, V, o) be the coordinates of one these points T, then 



. t 



pi = X + - 



pr} = y + 



hj 



(i; 



in passing to points near to PP', i, y), ^, v, w remain iinchanged; 

 therefore, 



f*p = ^-fi-:^U??. (2) 



ax' / ax 



with four similar equations in y, %, v, w, from which, by putting for 

 ^, f], t, . . ., their values from (1), we easily get 



tiX 



bp x{ax-'+e) _ xh9 ^f^^'^_ A 



ax? — 6 ax- - 



bp ?/{h'+0) yt>0 

 by = — 



p h' - ^ h^ 



26 



P 



X X'Qp 



■7, V 



J ax- - p 



y ^y^p 



h/-6 



(3) 



and since 



x-\-y + %-\-v-\-w = 0, b*' + by + bs + bv + bti) = 0, 



we have for 6 the equation 



% V w 



X y 



+ 



+ 



= 0, 



{AY 



ax- -6 hf- 6 cz- -6 dv"- 

 which we shall write 



ax^ -$ ' ■ ■ 



It contains the irrelevant factor 6 ; dividing by this, there remains a 

 quadratic. 



Denoting the roots of this qiiadratic by 6i, 6^, we see that the line 

 PP' touches C in two points, Tand T', whose coordinates are given 

 by putting 0^ and O-. for ^ in (1). 



1 It is easy to see that this equation may be written 



*■ =0, where A, B, C are any three quantities. 



ax (ax"^ —6) 

 In fact, I, m, oi can he found, so that 



Aa-x* + Box"- + C 



2 7-2—^— - 2 Ua; + — + . 



ax iax' — 6) \ ax ax'' 



I in nx \ 



\ ax ax — e/ 



(5) 



