594 



Proceedings of the Royal Irish Academy. 



LXXX. — OiN'iHi; Equatiost of a Takgenx Coxe to a Quabeic, eekeeeid 

 TO THE Axes. Ey Job:n- C. Malet, Professor of Mathematics, 

 Queen's College, Cork. 



[Eead, Apidl 10, 1882.] 



The following purely analytical method of forming this equation is I 

 believe new : — 

 If the quadiic 



ax^ + 5y^ + c%^ + 2 hxy + 2gxz + 2/ys -\- 2lx ■{■ 2my + 2n% + d = 



he referred to its axes, the equation is 



where 



x^ y^ %- A 



— V — -\ 1- — - 



a ^ /3 ^ y ' S ' 



a 





h 





9 





I 



h 



h 





f 



m 



9 



f 





e 



n 



I 



m 





n 



d 





a 





h 





9 







h 





h 





f 







9 





f 





c 





s = 



and a, y8, y are the roots of the cubic equation in A formed by equat- 

 ing to the discriminant of 



ax^ + hy'^ + c%^ + 2hxy + 2gxti + 2fyz 



y^ 



X 



= 0. 



If the quadric be a cone, A vanishes ; and in this case being only con- 

 cerned with the ratios of a, (S, y, we may write - for -, where ^ is a 



A A 



constant selected at will. 



Consider now the equation of the tangent cone to the quadiic 



VIZ. — 



where 



and 



X' 



a' 





J_ _ 



1-0, 







SS'- 



-p^ = 



0, 





S' 



_x'^ 

 ~ a" 





^'2 



1, 



F 



xx' 



., yy' 





- 1 



