76 THE THEORY OF SCREWS. [85, 



the same as those of the original equation. The required theorem will 

 therefore be proved if it can be shown that all the roots of this equation 

 are real. That this is so is shown in Salmon s Modern Higher Algebra, 

 Lesson VI.* 



86. Another investigation of the Principal Screws of Inertia. 



The n Principal Screws of Inertia can also be investigated in the following 

 fundamental manner by the help of Lagrange s equations of motion in 

 generalized co-ordinates. 



Let . . . be the co-ordinates ( 95) of the impulsive screw. Let 

 &amp;lt;/&amp;gt;!,... &amp;lt; n be the co-ordinates of the body, then &amp;lt;j&amp;gt;i, ... &amp;lt;f&amp;gt; n will be the co-ordinates 

 of the instantaneous screw, and from Lagrange s equations, 



_^ = 

 dtdfj d$r 



where T is the kinetic energy and where P-$$i denotes the work done in a 

 twist S&amp;lt; t against the wrench. 



If the screws of reference be co-reciprocal and if &quot; be the intensity of 

 a wrench on , then 



p, - 2^r&. 



As we are considering the action of only an impulsive wrench the effect of 

 which is to generate a finite velocity in an infinitely small time we must 

 have the acceleration infinitely great while the wrench is in action. The term 



, - is therefore negligible in comparison with - ( r ) and hence for the 

 dfa dt \d&amp;lt;j)J 



impulsive motion -f 



d (dT\ 



*Wr?S 



We may regard i and &quot; as both constant during the indefinitely small time 

 e of operation of the impulsive wrench, whence ( 79) 



2f 1 f&quot; = ^. 

 Pi cty, 



Hence replacing &amp;lt;fa l} ... (j&amp;gt; n by 6 lt ... B n we deduce the following ( 95, 96). 



If T be the kinetic energy of a body ivith freedom of the nth order, 

 twisting about a screw 6 whose co-ordinates referred to any n co-reciprocals 

 belonging to the system expressing the freedom are B l} ... n , then the co-ordinates 



* See also Williamson and Tarleton s Dynamics, 2nd edition, p. 457 (1889), and Kouth s 

 Rigid Dynamics, Part II, p. 49 (1892). 



t Niven, Messenger of Math., May 1867, quoted by Bouth, Rigid Dynamics, Part I, pp. 327-8. 



