198 Mr Dixon, On a method of discussing [May 25. 



(6) On a method of discussing the plane sections of surfaces. 

 By A. C. Dixon, M.A., Trinity College. 



The plane sections of surfaces are generally treated by examin- 

 ing their projections upon one of the co-ordinate planes. It is 

 possible however to find at once the equation to any section of 

 any surface referred to any axes, or indeed to any triangle, in its 

 plane. 



Let (#!, y lt z 1 )(x 2 , y 2 , z 2 )(x 3 , y z , z 3 ) be the co-ordinates of the 

 three vertices of the proposed triangle of reference referred to 

 the original axes. Let (u, v, w) be the areal co-ordinates of any 

 point in the plane of the triangle referred to that triangle. Then 

 the co-ordinates of the same point referred to the original axes are 



{ux 1 -\-vx 2 J rWx 3 , uyi + vy 2 + wy s , uz-y\-vz 2 -\-wz 3 ). 



By substituting these for x, y, z in the equation to any surface 

 we have the equation, in areal co-ordinates, to the section of the 

 surface made by the plane of the triangle. 



As an application — if we write down the condition that this 

 curve should have a double point and in this condition suppose 

 #3 , yz, z 3 to be current co-ordinates we have an equation which 

 will represent the tangent planes that can be drawn to the surface 

 through the straight line joining (x 1} y 1} z^ to (x 2) y 2 , z 2 ). 



Thus for a conicoid we have the equation to a pair of tangent 

 planes in the form 



S, S 1} S 2 = 0, 



s 2 , 



$ = being the equation to the surface itself, and 



(S, Sn, S 22 , S 12 , S 2 , &$w, u, v) 2 = 



the areal equation to the section. 



If we wish to have Cartesian instead of areal co-ordinates in 

 the plane, suppose (f , rj, £) to be the origin and (l lt m l5 ^i 1 )(4, m 2 , n 2 ) 

 the direction-cosines of the axes, rectangular or oblique. Denote 

 the co-ordinates in the plane by (u, v) and the co-ordinates of the 

 same point referred to the old axes are 



(f + ku + Lv, 7} + rant, + m 2 v, £ + n x u + n 2 v). 



For instance, let us consider a plane section of the surface 



ax 2 + by 2 + cz 2 = 1 . 



