POTENTIAL ENERGY OF A DISPLACEMENT. 69 



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. The values of r substituted successively in the linear 



a a 



equations just written will determine the co-ordinates 

 of the n principal screws of the potential. 



We can now show that these n screws are co-reci- 

 procal. It is evident, in the first place, that if S be 

 a principal screw of the potential, and if be a displace- 

 ment screw which evokes a wrench on ?j, the principle 

 of 70 asserts that, when is reciprocal to S, then must 

 also rj be reciprocal to S. Now, let the n principal 

 screws of the potential be denoted by Si, . . . , S n , and let 

 T n be that screw of the screw complex which is recipro- 

 cal to S ly . . . , , S n . i ( 65), then if the body be displaced 

 by a twist about T n , the wrench evoked must be on a 

 screw reciprocal to Si, . . . . , S n . i ; but T n is the only 

 screw of the screw complex possessing this property; 

 therefore a twist about T n must evoke a wrench on T n , 

 and therefore T n must be a principal screw of the poten- 

 tial. But there are only n principal screws of the 

 potential, therefore T n must coincide with S n , and there- 

 fore S n must be reciprocal to Si, . . . . S n - 1- 



72. Co-ordinates of the "Wrench evoked by a Twist. 

 The work done in giving the body a twist of small am- 

 plitude a about a screw a, may be denoted by 



In fact, remembering that Q'OI = a/, . . . , and substituting 

 these values for a/ in F( 70), we deduce the expression : 



f . . . + A nn a n z + 2A lz a l a 2 + 2 A 13 a!a 3 + . . . 



where F is a certain constant, whose dimensions are a 

 mass divided by the square of a time, and where v a is a 

 linear magnitude specially appropriate to each screw a, and 



