394 THE THEORY OF SCREWS. [356, 



where (11), (12), &c., are n? coefficients depending on the distribution of the 

 masses, and the other circumstances of the mass-chain and its constraints. 

 The equations having this form, the necessary one-to-one correspondence is 

 manifestly observed. 



357. The principal screw-chains of Inertia. 



We are now in a position to obtain a result of no little interest. Just as 

 we have two double points in two homographic rows on a line, so we have n 

 double chains in the two homographic chain systems. If we make, in the 

 foregoing equations, 



0i = p*i 5 #2 = p 2 , &c., 



we obtain, by elimination of a 1} ... a n , an equation of the nth degree in p. 

 The roots of this equation are n in number, and each root substituted in the 

 equations will enable the co-ordinates of each of the n double screw-chains 

 to be discovered. The mechanical property of these double chains is to be 

 found in the following statement : 



If any mass-chain have n degrees of freedom, then in general n screw- 

 chains can always be found (but not more than n), such that if the mass-chain 

 receive an impulsive wrench from any one of these screw-chains, it will 

 immediately commence to move by twisting about the same screw-chain. 



In the case where the mass-chain reduces to a single rigid body, free or 

 constrained, the n screw-chains to which we have just been conducted reduce 

 to what we have called the n principal screws of inertia. In the case, still 

 more specialized, of a rigid body only free to rotate around a point, the 

 theorem degenerates to the well-known property of the principal axes. We 

 may thus regard the n principal chains now found as the generalization of 

 the familiar property of the principal axes for any system anyhow con 

 strained. 



Considerable simplification is introduced into the equations when, instead 

 of choosing the chains of reference arbitrarily, we select the n principal 

 screw-chains for this purpose ; we then have the very simple results, 



0, = (11) x ; 2 = (22) 2 ; ... 6 n = (nn) a n . 



This gives a method of finding the impulsive screw-chain corresponding to 

 any instantaneous screw-chain. It is only necessary to multiply the co 

 ordinates of the instantaneous screw-chain i, a 2 by the constant factors (11), 

 (12), &c., in order to find the co-ordinates of the impulsive screw-chain. 



The general type of homography here indicated has to be somewhat 

 specialized for the case of impulsive screw-chains and instantaneous screw- 

 chains. The n double screw-chains are generally quite unconnected we 

 might, indeed, have exhibited the relation between the two homographic 



