78 THE THEORY OF SCREWS. [86- 



Hence we find that for all screws on the cylindroid represented by 6 l} 2 , 0, 

 the energy will vary as the pitch when the twist velocity remains the same. 

 It appears from the representation of the Dynamical problem in chap. XII. 

 that in this case all the screws of the cylindroid O lt 2 , 0, must be principal 

 screws of Inertia. The number of principal screws of inertia is therefore 

 infinite in this case. (See Routh s Theorem, Appendix, Note 2.) 



87. Enumeration of Constants. 



It is the object of this article to show that there are sufficient constants 

 available to permit us to select from the screw system of the ?ith order 

 expressing the freedom of a rigid body, one group of n screws, of which every 

 pair are both conjugate and reciprocal, and that these constitute the principal 

 screws of inertia ( 78). 



To prove this, it is sufficient to show that when half the available con 

 stants have been disposed of in making the n screws conjugate (81) the 

 other half admit of adjustment so as to make the screws also co-reciprocal. 

 Choose A! reciprocal to B lt ... B 6 _ n , with n - 1 arbitrary quantities; A z 

 reciprocal to A ly B^, ... B K _ n&amp;gt; with n 2 arbitrary quantities, and so on, then 

 the total number of arbitrary quantities in the choice of n co-reciprocal 

 screws from a system of the ?ith order is 



n(n 1) 

 n-l + w-2... +1= -y . 



Hence, by suitable disposition of the n(n 1) constants it might be 

 anticipated that we can find at least one group of n screws which are 

 both conjugate and co-reciprocal. 



We have now to show that these screws would be the principal screws 

 of inertia ( 78). We shall state the argument for the freedom of the third 

 order, the argument for any other order being precisely similar. 



Let AI, A 2 , A 3 be the three conjugate and co-reciprocal screws which 

 can be selected from a system of the third order. Let B lt B. 2 , B 3 be any 

 three screws belonging to the reciprocal screw system. Let R lt R y , R. } be 

 any three impulsive screws corresponding respectively to A l} A z , A 2 as 

 instantaneous screws. 



An impulsive wrench on any screw belonging to the screw system of the 

 4th order defined by R lt B l} B z , B 3 will make the body twist about A, ( 82), 

 but the screws of such a system are reciprocal to A 2 and A a ; for since A l and 

 A z are conjugate, R l must be reciprocal to A z ( 81), and also to A 3 , since A^ 

 and A 3 are conjugate. It follows from this that an impulsive wrench on any 

 screw reciprocal to A 2 and A 3 will make the body commence to twist about 

 A lt but A l is itself reciprocal to A 3 and A a , and hence an impulsive wrench 



