238 VI. SYSTEMS OF VECTOES. DISTBJBUT. OF MASS. INSTANT. MOTION. 



foot of the perpendicular from 6r. Then by the two theorems of 

 70 and 71, 



K = Ace 2 + BP* + Cf + 



fc 2 = a 2 a 2 + 6 2 /3 2 + c 2 y 2 -f p\ 

 Now 



tf = r 2 - (f = r 2 - (ax + /3y + r*) 2 , 



94) &2 = ^2 + 202 + c2;; 2 _j_ ^2 _ q 2 



In order to find the principal axes at we must make this a 

 maximum or minimum with respect to a, ft y subject to the condition, 



a 2 + /3 2 -f r 2 = I- 



Multiplying this by a constant tf, subtracting from 94), and diffe- 

 rentiating 



95 ) {^ 2 - <5 ( 



Multiplying these equations respectively by , ft y and adding, 



a 2 a 2 + tfp + cV - g (aa; + fty + ye) - (3 = 0, 



fc 2 _ r 2 _ 6 = 0< 



Thus tf is determined as 



97) 6 = W- r 2 . 

 Inserting this value in 95) we have 



(a 2 + r 2 - F) a = qx, 



98) (& 2 + r 2 - F) /3 = 2 y, 

 (c 2 +r a -fc 8 )y =g^. 



Multiplying these equations respectively by 



x y e 



and adding, we get, since q divides out, 



" " 4- z * - 1 



-- - 



If we now put r 2 & 2 = ^), this is the same cubic as 83) to 

 determine 9, and gives three real roots for & 2 , 



7. 2 _ 0.2 1 



/fcj T ~ A , 



