68 POTENTIAL ENERGY OF A DISPLACEMENT. 



the screws reciprocal to 7 - n screws constitute a screw 

 complex of the (n- i) th order ( 46). 



The reader will be careful to observe the distinction 

 between two conjugate screws of inertia ( 54), and two 

 conjugate screws of the potential. Though these pairs 

 possess some useful analogies, yet it should be borne in 

 mind that the former are purely intrinsic to the rigid 

 body, inasmuch as they only depend on the distribu- 

 tion of its material, while the latter involve extrinsic 

 considerations, arising from the forces to which the 

 body is submitted. 



71. Principal Screws of the Potential. We are now 

 going to prove that n screws can be found such that when 

 the body is displaced by a twist about any one of these 

 screws, a reduced wrench is evoked about the same 

 screw. The screws which possess this property are called 

 the principal screws of the potential. Let a be a principal 

 screw of the potential, then we must have, 69 : 



01 ' - 



and (n - i) similar equations. 



Introducing the value of V a> and remembering (36) 

 that a/' = a"cti and a/ = a'ai, we have the following n 

 equations : 



/ " \ 



qifAu + -7^1 j + a 2 A lz + . . . +a n Am = o, 



&c., &c. 



A (A " \ 



+ a z A nz + . . + a n ( Ann + p = O. 



\ o / 



From these linear equations Ql , a n can be elimi- 

 nated, and we obtain an equation of the n th degree* in 



* All the roots of this equation are real. See Salmon's Higher Algebra, 

 Art. dd. 



