IV. NOTE OX SURFACES OF THE SECOND ORDER. 



[Proceedings of the Royal Irish Academy, VOL. in. p. 429. Read April 12, 1847.] 



LET a surface A of the second order be represented by the 

 equation 



fl l! = 

 T + Q + -Ko~ ' 



its primary axis being that of x. Through a given point 8 

 whose co-ordinates are #', y', z', conceive three surfaces confocal 

 with A to be described, and let P, P', P", be the squares of 

 their primary semiaxes. Then if normals drawn to these sur- 

 faces respectively at the point 8 be the axes of a new system of 

 co-ordinates , TJ, , and if we put 



~'2 ,/2 -'2 



P_ p _ z. p' _ p _ i.' p" p _ i." , y , f 

 L o - > * -co --> 4 ^o-rt- "B" + 7Tjy ' 



x^o ^o -to 



the equation of the surface A, referred to the new co-ordinates, 

 will be 



where ^ , ?o, Zo are the co-ordinates of its centre. 



From the form of this equation it is evident that, if the sur- 

 face be intersected by the plane whose equation is 



