66 POTENTIAL ENERGY OF A DISPLACEMENT. 



with respect to the corresponding amplitude, and chang- 

 ing 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 6, are proportional to 



(67) 



2p l d'0 l '' ' 2p n dti n ' 



| 70. Conjugate Screws of the Potential. Suppose that 

 a twist about a screw 9 evokes a wrench on a screw 77, 

 while a twist about a screw evokes a wrench on a 

 screw . If 9 be reciprocal to , then must be reciprocal 

 to r\. This will now be proved. 



The condition that 9 and are reciprocal is 



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 



UW +....+ p n 9 n 'Zn" = o ; 

 but we have seen ( 69) that 



whence the condition that 9 and are reciprocal is 

 0> dV * * ^> dV * 



01 -rf + + V n -= = O. 



4f^ a^,' 



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

 order of the quantities $\, . . . , n x , we may write 



