230 THE THEORY OF SCREWS. [219- 



the six absolute principal screws of inertia as screws of reference, ( 79) then 

 the body will commence to move as if it were free, but had been acted upon 

 by a wrench of which the co-ordinates are proportional to p^, ..., p 6 6 6 . 

 It follows that the given impulsive wrench, when compounded with the 

 reactions of the constraints, must constitute the wrench of which the co 

 ordinates have been just written ; whence if h be a certain quantity which 

 is the same for each co-ordinate, we have the six equations 



Multiply the first of these equations by X 1; the second by Xj, &c.: adding 

 the six equations thus obtained, and observing that 6 is reciprocal to X, and 

 that consequently 



2M\i = 0, 

 we obtain 



f} / &quot; ^r} 1 \ 1 + X &quot;^! 2 + //&quot;SXj//,! = 0, 



and similarly multiplying the original equations by p 1 , ..., ^ 6 and adding, 

 we obtain 



^x, + /&quot;S 2 = 0. 



From these two equations the unknown quantities X &quot;, //&quot; can be found, 

 and thus the initial reaction of the constraints is known. Substituting the 

 values of X &quot;, //&quot; in the six original equations, the co-ordinates of the 

 required screw 9 are determined. 



220. Principal Screws of Inertia in the Four-System. 



We have already given in Chapter VII. the general methods of deter 

 mining the principal screws of inertia in an ?i-system. The following is a 

 different process which though of general application is in this chapter set 

 down for the case of the four-system. 



Choose four co-reciprocal screws a, ft, 7, 8 of the four-system and let 

 their co-ordinates be as usual !,... ,fc; &,...,&; y n ...,7; 81, ...,8 6 ; 

 referred to the six absolute principal screws of inertia ( 79). 



Let an impulsive wrench on one of the principal screws of inertia 6 in 

 the four-system be decomposed into components on a, /3, 7, 8, and let the 

 impulsive intensities be a &quot;, &quot;, 7 &quot;, 8 &quot;. 



Let X, p, be any two screws on the reciprocal cylindroid. Then the body 

 will move as if it had been free and had received impulsive wrenches on the 

 absolute principal screws of inertia, the impulsive intensities being 



a&quot; a 6 + {3&quot; j3 G + y &quot; 7ti + B &quot;S V + X /7/ X 6 



