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IV. NOTE ON SURFACES OF THE SECOND ORDER. 



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



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

 equation 



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

 whose co-ordinates are a?', /', s', 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 S be the axes of a new system of 

 co-ordinates ?, ?j, , and if we put 



P-P,-*, p'-p,"*', p"-p,-r, J + ^ + ^ /, 



jfo ^0 *H> 



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

 will be 



where 5 , no, So 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 



