100] THE POTENTIAL. 89 



If a free or constrained rigid body be displaced from a position of equi 

 librium by twists of small amplitudes, #, , ... # , about n co-reciprocal screws 

 of reference, and if V denote the work done in producing this movement, 

 then the reduced wrench has, for components on the screws of reference, 

 wrenches of which the intensities are found by dividing twice the pitch of 

 the corresponding reference screw into the differential coefficient of V with 

 respect to the corresponding amplitude, and changing the sign of the 

 quotient. 



It is here interesting to notice that the co-ordinates of the reduced 

 impulsive wrench referred to the principal screws of inertia, which would 

 give the body a kinetic energy T on the screw 0, are proportional to 



^dT 1 dT 

 2 Pl de, &quot;2p n d0 n vs 



100. Conjugate Screws of the Potential. 



Suppose that a twist about a screw 6 evokes a wrench on a screw ?/, 

 while a twist about a screw &amp;lt;f&amp;gt; evokes a wrench on a screw . If 6 be reci 

 procal to , then must &amp;lt;f&amp;gt; be reciprocal to ij. This will now be proved. 



The condition that 6 and are reciprocal is 



pAti + + p n 6 n n = ; 



but the intensities (or amplitudes) of the components of a wrench (or twist) 

 are proportional to the co-ordinates of the screw on which the wrench (or 

 twist) acts, whence the last equation may be written 



but we have seen ( 99) that 



_ 1^ t _ 



*%V&amp;lt;*fc&quot; nn 



whence the condition that 9 and f are reciprocal is 



0/jEj^p9W&** 



d(f)i d&amp;lt;f) n 



Now, as V$ is an homogeneous function of the second order of the quantities 

 &amp;lt;J&amp;gt;i, ... (j&amp;gt; n } we may write 



V* = A u ft* + . . . + A nn $n&amp;gt; + 2^ 12 &amp;lt;k&amp;gt; s + 2^/0; + . . . , 



in which A hk = A ai . 

 Hence we obtain: 



