568 



NOTES. 



If we multiply this equation by an arbitrary constant /I and add it to 

 _F(a, |3, y), we obtain the condition by writing the derivatives of F lq> 

 equal to zero Thus we obtain 



4) 





Now the direction cosines of the normal to the quadric at a point #, y, & 

 are proportional to 



^E. d JL d JL. 



dx dy dz 



At points where the normal is in the direction of the radius vector we 

 have 



dF dF dF 



dx _ dy dz 



x y z 

 But 



dF(x, y, z) dF(cc, ft y) 



dx 



da 



etc., 



so that the equations 4) show that at the ends of the principal axes the 

 tangent plane is perpendicular to the radius vector. 



Effecting the differentiations the equations 4) become 



(A - 



5) 



F0 



Dfi 



7 =0, 



y =0, 

 -h(<7-/l)y=0. 



The condition that these equations, linear in a, |3, y, shall be compatible 

 for values of a, |3, y, other than zero is that the determinant of the 

 coefficients shall vanish. 



I, F , 

 F , B - A, 



E 

 D 



E 



D 



(7 



This is a cubic in A, which being expanded is 



