90 THE THEORY OF SCREWS. [100- 



Introducing these expressions we find, for the condition that and should 

 be reciprocal, 



This may be written in the form : 



.. /) i i i . A a &amp;gt; i i i f Q I i f) i &amp;gt;\ I A 



&quot;-ll* ! Y I i &quot; L nn&quot;n 0n T &quot;1J \&quot;l 0-2 T t Qi ) i = &quot; 



But this equation is symmetrical with respect to and &amp;lt;, and therefore 

 we should have been led to the same result by expressing the condition that 

 was reciprocal to 77. 



When and &amp;lt;/&amp;gt; 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 system of the fifth order, which embraces all the screws reciprocal to 

 77, as in a certain sense the locus of &amp;lt;. All the screws conjugate to 0, and 

 which, at the same time, belong to the screw system C by which the freedom 

 of the body is defined, must constitute in themselves a screw system of the 

 (n l)th order. For, besides fulfilling the 6 n conditions which define the 

 screw system C, they must also fulfil the condition of being reciprocal to 77 ; 

 but all the screws reciprocal to 7 n screws constitute a screw system of the 

 (ra-l)th order ( 72). 



The reader will be careful to observe the distinction between two conju 

 gate screws of inertia ( 81 ), and two conj ugate 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 distribution of its material, while the latter involve extrinsic 

 considerations, arising from the forces to which the body is submitted. 



101. Principal Screws of the Potential. 



We now prove that in general 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 on 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, 99: 



dV a 



i = + , , 

 2wj dflj 



and ( /i 1) similar equations. 



