86] THE PRINCIPAL SCREWS OF INERTIA. 77 



of an impulsive wrench by which the actual motion of the body could be 

 produced are proportional to 



dT ^dT_ 



p l d0 1 &quot; p n dd n 



The existence of n Principal Screws of Inertia can now be readily deduced, 

 for suppose that 



l dT l dT 



where X is an unknown factor. If then we make 



T = a u 0!~ + a&amp;lt;v&amp;gt;Q&amp;gt;? + 2,a K 0i0&amp;lt;&amp;gt; . . . 

 we have an equation of the nth degree for X as follows : 



tt-ia , . . . a^n =: &quot; 



Ct n i , Gt n2 &amp;gt; &nn 



It is essential to note that T is A function of such a character that by 

 linear transformation it can be expressed as the sum of n squares, for suppose 

 it could be expressed as 



it would be possible to find a real screw which made H 1} H 2 , ... II n -i each 

 zero, and then the kinetic energy of the body twisting about that screw 

 would be negative. Of course this is impossible. Hence we deduce from 

 85 the important principle that all the Principal Screws of Inertia are real. 



If the equation had a repeated root the number of Principal Screws of 

 Inertia is infinite. We take n = 4f, but the argument applies to 3 and 2 

 also. (There can be no repeated root when n is either 5 or 6. See chaps. 

 XVII. and XVIII.) We can choose variables such that T becomes 



and the pitch X becomes simultaneously 



If therefore the discriminant of T + \p, equated to zero, has a pair of equal 

 values for X, we must have a condition like 



Take any screw of the system for which ;i = 0, # 4 = 0, then 



T = M 6* (1*1*0!* + ii*0.?), 

 P= p 



, 



or T= p. 



Pl 



