486 THE THEORY OF SCREWS. 



In the case of n -- 4 the function T will consist of 10 terms such as 



If any arbitrary values be assigned to A u , A }a &c., it will still be possible to 

 determine a rigid body such that this function shall represent u e 2 (to a constant 

 factor), because we have 9 co-ordinates disposable in the rigid body. Hence for 

 n - 4 and a fortiori for any value of n less than four the function representing T 

 will be a function in which the coefficients are perfectly unrestricted. Hence 

 n = or &amp;lt; 4 the determinantal equation is in our theory of the most general type. 

 The general theory while affirming that all the roots are real does not prohibit 

 conditions arising under which roots are repeated. Hence Routh s important 

 theorem becomes of significance in cases n=2, n 3, n = 4: for in these equations 

 the roots may be repeated. 



But in the case of n - 5 the function T consists of 1 5 terms. If arbitrary 

 values could be assigned to the coefficients then of course the general theory would 

 apply and cases of repeated roots might arise. But in our investigation the 15 

 coefficients are functions of the co-ordinates which express the most general place 

 of a rigid body, and these co-ordinates are not more than nine. If these nine co 

 ordinates were eliminated we should have five conditions which must be satisfied 

 by the coefficients of a general function before it could represent the T of our 

 theory even to within a factor. The necessity that the coefficient of T shall satisfy 

 these equations imports certain restrictions into the general theory of the deter 

 minantal equation based on T. One of these restrictions is that T shall have no 

 repeated roots. The same conclusion applies a fortiori to the case of n = 6. 



The subject may also be considered as follows. 



Let us first take the general theorem that when reference is made to n 

 principal screws of inertia of an n-system the co-ordinates of the impulsive wrench 

 corresponding to the instantaneous screw 



are ( 97) 



^a,.-- X 



Pi Pn 

 For a principal screw of inertia the ratios must be severally equal or 



Pi &amp;lt;*i Fa &quot;2 Pn a-n 



These equations can generally be only satisfied if n - 1 of the quantities 



be zero, i.e. there are in general no more than the n principal screws of inertia. 

 If however 



Pi P-2 



then though we must have 



a, = 0...a w = 0, 

 aj and a 2 remain arbitrary. 



