236 VI. SYSTEMS OF VECTORS. DISTRIBUT. OF MASS. INSTANT. MOTION. 



Call the roots in order of magnitude A, ^, v. The changes of 

 sign above show that I lies in the interval A > c 2 necessary in 

 order that the surface shall be an ellipsoid, ^ in the interval 

 - c 2 > ^ > fr 2 that it may be an hyperboloid of one sheet, and v 

 in the interval fr 2 > v > a? that it may be an hyperboloid of 

 two sheets. There pass therefore through every point in space one 

 surface of each of the three kinds. If we call 



85) F(l, x, y, ,) = -^ + -^ + -^ - 1, 



the equation F=Q defines A as a function of x,y,8. The normal 

 to the surface A (xyz) = const, has direction cosines proportional to 



til dl dl 



o * o y ^^' 

 ex cy cz 



Now since identically _F = 0, differentiating totally, 



3F , dF , dF , . 0.F , . 



and we have 



dF 



for the required partial derivative of A with respect to #, when 

 and are constant. 

 Therefore 



31 2x I i x* * z* 2 a; 



Similarly 



86 ) 



dz 



The sum of the squares of the derivatives being called /^ 2 , we have 

 4 



Now the direction cosines of the normal to the surface A = const, are 



cos (n*x) = T- o = 

 88) cos (n^y) = + 



cos 



