THE PRINCIPAL SCREWS OF INERTIA. 5 1 



quantities in the choice of n co-reciprocal screws from a 

 complex of the n lh order is 



n(n- i) 



n i+n z...+ i= 



2 



Hence, by suitable disposition of the n(n - i) constants 

 we can find one group of n screws which are both con- 

 jugate and co-reciprocal. 



We have now to show that these n screws are really 

 the principal screws of inertia (51). yWe shall state the 

 argument for the freedom of the third order, the argu- 

 ment for any other order being precisely similar. 



Let A i, A 2, A z , be the three conjugate and co-reci- 

 procal screws which can be selected from a complex of 

 the third order. Let B^B^ Bj, be any three screws belong- 

 ing to the reciprocal screw-complex. Let R l9 jR 2 , R* 

 be any three impulsive screws corresponding respectively 

 to A l9 A 2 , A 3 as instantaneous screws. 



An impulsive wrench on any screw belonging to the 

 screw-complex of the 4* order defined by Ri 9 B l9 B 2 , B* 

 will make the body twist about A\ (55), but the screws 

 of such a complex are reciprocal to A z and A 3 ; for since 

 A i and A z are conjugate, ^ must be reciprocal to A z 

 ( 54), and also to A 3 , since AI and Az are conjugate. It 

 follows from this that an impulsive wrench on any screw 

 reciprocal to A z and A 3 will make the body commence 

 to twist about A 19 but A 1 is itself reciprocal to A z and A 3 , 

 and hence an impulsive wrench on AI will make the 

 body commence to twist about AI. Hence AI and also 

 At and A 3 are principal screws of inertia. 



We shall now show that with the exception of the n 

 screws here determined, no other screw possesses the 

 property in question. Suppose another screw S were to 

 possess this property. Decompose the wrench on S into 

 n wrenches of intensities Si", &c. S n "onA lf &c. A, this 

 must be possible, because if the body is to be capable of 



E 2 



