440 Focal Trojierties\of Surfaces of the Second Order. 



meeting one of the directrix planes in a right line, and the 

 tangents to the lines of curvature be produced to meet this 

 line in the points m, m', the sphere described on m m 1 as dia- 

 meter will pass through the point of contact and the focal 

 centre ; hence may be given a new method of determining 

 the lines of curvature. 



Prop. IX. — Through the focal centre of a surface (X), let 

 a right line and a plane be drawn perpendicular to each 

 other -, the line meeting the surface in a point t, and the 

 plane meeting one of the directrix planes in a right line m m! '; 

 the plane mm' t envelopes a surface of revolution, whose focus 

 is the focal centre of (2), and whose directrix plane passes 

 through the intersection of the conjugate directrix planes of 

 the given surface. 



Let the equation of the surface, the origin being placed at 

 the focal centre, be 



x n q n z i2 2 e x f V 1 



-gr + V + . -ir r-TW'W «0 being the 



coordinates of the point r; let £ = ju, ? + a, (b) v = v ?+/3, 

 (b') be the* tangential equations of the line mm! in the direc- 

 trix plane ; and x' J -f j/ v + z 1 ? = 1 (c) the equation of the 

 plane passing through the point [x 1 ' y ] V) and the right line 

 m m\ 



In- the first place, as the line m m l is in a plane passing 



X If 



through the origin, \l — — , v = ~ ; and equations (b) (b') 



z z 



x' y' 



are changed into £ = — ? + a, y = ~? + /3; and as the 



z z 



line m m' is in the directrix place of which the tangential co- 

 ordinates are 



£ = J- , u = 0, ? = -£-- ; these values of £, u, £ 



must satisfy the tangential equation of the right line m m ; by 

 these "substitutions the equations (b) (b') are changed into 



z'c(&t+ae) = J(b*ct-qbri)\ (f] , 

 z'cb°~v =^(b i cg-abti)J * } 



and we have now to eliminate x l if z l between the four equa- 

 tions (a) (c) (d) ; from the three latter we get 



6 9 c 2 £— ac"e 



j __ 



Z> 2 c 2 (£ 2 + u 2 + $ 2 ) +ad*e£-abcit 



, l> 2 c°- o 



y ~" b*(?{P + «* + ?*) + ac 2 e£-abcYi$ 

 * See the treatise quoted above, page 11. 



