102] THE POTENTIAL. 91 



Introducing the value of V a , and remembering (34) that a/ = &amp;lt;*&quot;! and 

 a/ = a ci, we have the following n equations: 



(ft \ 

 AH rpi) + a 2 A 12 + ... + a n A ln =0, 

 a / 



&c., &c. 

 aiA m + a. 2 A n2 + ... +a n (A nn T p n j = 0. 



\ ^* / 



From these linear equations a n ... a n can be eliminated, and we obtain 



// &quot; 



an equation of the nth degree in . The values of substituted suc 

 cessively in the linear equations just written will determine the co-ordinates 

 of the n principal screws of the potential. If the position of equilibrium 

 be one which is stable for all displacements then V a must under all 

 circumstances be positive. As it can be reduced to the sum of n squares 

 all the roots of this equation will be real ( 86) and consequently all the n 

 principal screws of the potential will be real. 



We can now show that these n screws are co-reciprocal. It is evident, 

 in the first place, that if S be a principal screw of the potential, and if be 

 a displacement screw which evokes a wrench on 77, the principle of 100 

 asserts that, when 6 is reciprocal to S, then must also y be reciprocal to S. 

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

 T n be that screw of the screw system which is reciprocal to Si, ... S n -i ( 95), 

 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 system 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 

 potential. But there are only n principal screws of the potential, therefore 

 T n must coincide with S n , and therefore S n must be reciprocal to Si, ... S n ^. 



102. Co-ordinates of the Wrench evoked by a Twist. 



The work done in giving the body a twist of small amplitude a! about a 

 screw a, may be denoted by 



fttfta*. 



In fact, remembering that a a 1 = 1 , ... , and substituting these values for 

 0.1 in I 7 &quot; ( 100), we deduce the expression: 



Fv a - = A n a,- + ...+ A nn a n - + 24 18 * 1 a + 24 13 a, + . . . 



where F is independent of a and has for its dimensions a mass divided by the 

 square of a time, and where v a is a linear magnitude specially appropriate 

 to each screw a, and depending upon the co-ordinates of a, and the constants 

 in the function 



