Malet — On the Equation of a Tangent Cone to a Quadric. 595 

 To refer this cone to the axes, equate to the discriminant of 



A 



where ij~ ^\ ^' ^' 



a^ 0^ c- 



„ xx' yy' %%' 



d^ V- (? 



Differentiating {A) with respect to x, y, and %, we get the three equa- 

 tions 



^, ( I \\ x' ^ 



>S'^ ----- n - 0, 



\a- A] a- 



V c a) c~ 

 To eliminate x, y, % from these equations, multiply respectively by, 



■^ and 



a2_A' ¥-X' C--A' 



add and divide by 11, when we find 



V«-(rr - A) b~{o- - A) c\r - A) j 



or'"- «'2 s'^ 



a^ - A o" - A c" - A 



But the roots of this equation are cC' - a{-, cC" - ai, cC- - a^, where 



«i, «2, ^i^s are the semiaxes major of the confocals through x' , y', %' ; 



hence the equation of the tangent cone through this point is referred 



to the axes, 



x"^ xP- £" 



+ ~ — ^ + -. ^ = 0. 



a'' - a-c a~ — a^ a- - a^ 



In a similar manner the equation of the tangent cone through x' , y', z' 

 to the paraboloid 



z + jf ^'-^' 



when referred to the axes, is 



X^ 11- %- 



+ T -r + T T == 0, 



where Xj, Z3, Z3 are the values of L for the confocal paraboloids 

 through the point x', y', %'. 



R. I. A. PROC, SEE. II,, VOL. III. — SCIENCE. 3 F 



