POTENTIAL ENERGY OF A DISPLACEMENT. 67 



in which AM *= AI&. 

 Hence we obtain 



-^~ = 2 f Aufi + Anfa' + ---- 4 A ln $* 



Introducing these expressions we find, for the condition 

 that and should be reciprocal. 



0/(^n0i'+ . A ln <t>n') +....+ 0n f (A n A f + ....+ ^nn0) = O. 



This may be written in the form : 

 -4 aft V + . , A nn O n ' ft ' + -4 M (0iV + 0/00 + ....- o. 



But this equation \s perfectly symmetrical vntih respect 

 to 9 and 0, and therefore we should have been led to 

 the same result by expressing the condition that was 

 reciprocal to j. 



When and possess this property, they are said to 

 be conjugate screws of the potential, and the condition that 

 they should be so related, expressed in terms of their 

 co-ordinates, is obtained by omitting the accents from 

 the last equation. 



If a screw be reciprocal to 77, then is a conjugate 

 screw of the potential to 0. If we consider the screw 

 to be given, we may regard the screw complex of the 

 fifth order, which embraces all the [screws reciprocal to 

 aj, as in a certain sense the locus of 0. All the screws 

 conjugate to 0, and which, at the same time, belong to 

 the screw complex C by which the freedom of the body 

 is defined, must constitute in themselves a screw com- 

 plex of the (n - i ) th order. For, besides fulfilling the 6-n 

 conditions which define the screw complex C, they must 

 also fulfil the condition of being reciprocal to ij ; but all 



F 2 



