570 NOTES. 



Let us suppose a > b > c. and take the upper sign in 14). Then for 



I = oo f(l] is 4- 



c 4- 

 & 



o + 



00 



and the function f(l) behaves as shown in Fig. 13. As there are three 

 changes of sign, there are three real roots. It is to be noticed that the 

 reality of all the roots depends on #, r, s being of the same sign. Let 

 us call the roots A 15 A 2 , A 3 . Either one of these being inserted in the 

 equations 5), the equations become compatible, and suffice to determine 

 the ratios of the direction cosines. There are therefore always three 

 principal axes to a central quadric surface. If we call the cosines 

 belonging to the roots A x , a 1? /3 1? y 1? those belonging to A 2 , 2 , /3 2 , y 2 , 

 equations 5) become 



15) 



Multiplying the first three respectively by a 2 , /3 2 , y t and adding, 



4- 



, 

 4- D(ft yi 4- ft 72 4- ^y 2 ^ + yi 2 4- 



If we multiply the second three equations respectively by a 1? jSj, 7l and 

 add we obtain for 



the same expression. Accordingly we have 



17) (^ - A 2 ) (X a, 4- ft ft + 7l 72 ) = 0, 



so that if the roots /Lj, ^ 2 are unequal the corresponding axes are 

 perpendicular. In like manner if the determinantal cubic has three unequal 

 roots, the quadric has three mutually perpendicular principal axes. 



If two roots are equal the position of the corresponding axes becomes 

 indeterminate, and it may be shown that all radii perpendicular to the 

 direction given by the third root are principal axes of the same length. 

 The surface is then one of revolution about the determinate axis. If all 

 three roots are equal, the surface is a sphere, and any axis is a 

 principal axis. 



