THE CTTRYE ON ONE OF THE COORDINATE PLANES. 467 



We call, on the other band, 

 the central vectors of the points of contact, so that we have 



o-=e+tr 1 , 



§ ii- 



At the extremity of p x the tangent plane is by hypothesis the plane (j, k) 

 itself, so that the normal to the plane is parallel to i. At the same time, the 

 normal has the direction of cppi , thus we may put 



<pp =iE , 



N being a scalar to be determined. 



This equation gives 



p^lSip-H, 



where <£ _1 is the inverse of <£, and is defined by 



<p " 1 cd = aa?Saco + /36 2 S/3o> +jc 2 Sjw , 

 or briefly by 



Having 



Spi<f>P\ = ! > 



p t being a central vector of a point on the surface, we get by the two vector 

 equations : 



This determines N ambiguously, but we have the equation of the tangent plane 

 to give it its sign : namely, the tangent plane at the extremity of p x passes through 

 the point O ; we may, therefore, for the central vector o> put 



w = (-0). 



Thus the equation of the tangent plane 



Swq^ = 1 , 



owing to <f>pi = iN, becomes 



-NS0*=1. 

 This gives 



N: 



set j&fy-H 



If we put 



6 = iu+jv + hw , 



