103] 



THE POTENTIAL. 



93 



We now show that the expression of the potential will consist of the 

 sum of n square terms, when referred to any set of n conjugate screws of the 

 potential. 



The energy consumed in giving a body a twist of amplitude 6 from the 

 position of equilibrium to a new position P, is equal to Fv e *0 z ( 102), 

 and 77 is the screw on which the wrench is evoked. Suppose that from 

 the position P the body receive a twist of amplitude &amp;lt;j&amp;gt; about a screw &amp;lt;, it 

 would generally riot be correct to assert that the energy consumed in the 

 second twist was proportional to the square of its amplitude. For, during 

 the second twist, either a portion of the energy will be consumed in doing 

 work against the wrench on rj, or the energy expended in the second twist 

 will be rendered less, in consequence of the assistance afforded by the wrench 

 on t]. If, however, 77 be reciprocal to (f&amp;gt;, then the quantity of energy con 

 sumed in the twist about &amp;lt;/&amp;gt; will be unaffected by the presence of a wrench 

 on 77. Hence if 6 and &amp;lt;f&amp;gt; be two conjugate screws of the potential, the 

 energy expended in giving the body first a twist of amplitude 6 about 6, 

 and then a twist of amplitude &amp;lt;/&amp;gt; about &amp;lt;f&amp;gt;, is to be represented by 



By taking a third screw, conjugate to both 6 and &amp;lt;, and then a fourth 

 screw conjugate to the remaining three, and so on, we see finally that the 

 potential reduces to the sum of n square terms, where each pair of the 

 screws of reference are conjugate screws of the potential. 



