282] DEVELOPMENTS OF THE DYNAMICAL THEORY. 299 



To demonstrate the first of these formulae. Expand the left-hand side and 

 it becomes 



ft + &) (17, + 



i + ,) (ft + ft) + (a + ,) (ft + ft) + ( B + a.) (ft + ft)}. 



But, as already shown, 



cos (a?;) cos (a?;) 



whence, by substitution, the expression reduces to 



+ a (A + A) (i - 2 ) + a (! + a.,) (A - /3 2 ) 

 + 6 (A + ft) (. - 4 ) + 6 (a. + a 4 ) (/8, - /3 4 ) 

 + c (/3 6 + A) ( 5 - a e ) + c ( B + 6 ) (/3 5 - A) 



4 + 2ca 5 /? 5 - 



To prove the second formula it is only necessary to note that each side 

 reduces to 



It will be observed that these two theorems are quite independent of the 

 particular screws of reference which have been chosen. 



282. Conjugate Screws of Inertia. 



We have already made much use of the important principle that is 

 implied in the existence of conjugate screws of inertia. If a be reciprocal 

 to then must 17 be reciprocal to . This theorem implied the existence 

 of some formula connecting -nr a f and CT^. We see this formula to be 



Pa Pft 



cos ewj cos 



We have now to show that if ^^ = 0, then must w a f = 0. 



Let us endeavour to satisfy this equation when tn-^ is zero otherwise 

 than by making uj af zero. Let us make p a infinite, then -n^ will reduce 

 to %p a cos (af ) (for we may exclude the case in which p ( is also infinite 

 because in that case r af = 0, inasmuch as any two screws of infinite pitch 

 are necessarily reciprocal). 



