i66 Dr. J. BoTTOMLEV on 



From the two last equations we may deduce 



dz-- 

 hence we have 



(j^>-m>-' 



hence 



dZ dz dz dz 



dX^dx ^"^^ dY^'^ 



and from these equations we may obtain, 



the expression on the left measures the inclination of the 

 tangent plane at the point X, Y, Z, to the plane of xy, and 

 the expression on the right measures the inclination of the 

 tangent plane at x,y,s, to the same plane, hence these tan- 

 gent planes are parallel, therefore the line 



X-x _Y-y Z-z 

 c ~ c ~ c 



is normal to both surfaces. If a,j3, 7 be the direction angles 

 of the normal to the primitive surface 0(X, Y, Z,) we have 

 the equations 



vT = X - fCOSa 



J = Y - rcos/3 (20 



z = Z- <rcosy 



which may be written in the form 



