70 POTENTIAL ENERGY OF A DISPLACEMENT. 



depending upon the co-ordinates of a, and the constants in 

 the function F( 68). 



The parameter z> a may be constrasted with the para- 

 meter u a considered in 59. Each is a linear magni- 

 tude, but the latter depends only upon the co-ordi- 

 nates of a, and the distribution of the material of the 

 rigid body. Both quantities may be contrasted with 

 the pitch / a , which is also a linear magnitude, but 

 depends solely on the screw, and is therefore purely 

 geometrical. 



If a body receive a twist of small amplitude a' about 

 one of the principal screws of the potential, then the in- 

 tensity of the wrench evoked on the same screw is ( 69) : 



I * J / J 



2p a aa. 



but we have just seen that F= Fv^a*, whence we have 

 the following theorem : 



If a body which has freedom of the" n th order be 

 displaced from a position of equilibrium by a twist 

 about a screw a, of which the co-ordinates with respect 

 to the principal screws of the potential are ai, . . . . , a ny 

 then a wrench ( 66) is evoked on a screw of which 



the co-ordinates are proportional to ^-QI, . . . , a n , 



/l pn 



where v l9 &c., p ly &c., are the values of the quantity v, 

 and the pitch p, for the principal screws of the potential. 

 We can now express with great simplicity the con- 

 dition that two screws 9 and shall be conjugate screws 

 of the potential. For, if 9 be reciprocal to the screw 

 whose co-ordinates are proportional to 



^i 2 Vn Z 



fr 



we have : 



- O. 



