295] DEVELOPMENTS OF THE DYNAMICAL THEORY. 313 



and of X 



! cos + & sin , ... a 6 cos & + /3 6 sin &. 



In like manner, let p and p be two screws on a cylindroid, of which the 

 two principal screws are rj and f. 



Let &amp;lt;f&amp;gt;, &amp;lt; be the angles which p and p make respectively with . 

 Then the co-ordinates of p are 



r) l cos &amp;lt;/&amp;gt; + f j sin (j), ... rj 6 cos &amp;lt;/&amp;gt; + 6 sin &amp;lt;, 

 and of p 



7?! cos &amp;lt;/&amp;gt; + j sin &amp;lt;/&amp;gt; , ... 77,3 cos + 6 sin &amp;lt; , &c. 



We shall now suppose that the two cylindroids a, /3 and 77, are so 

 circumstanced that the latter is the locus of the impulsive wrenches cor 

 responding to the several instantaneous screws on the former with respect 

 to the rigid body which is to be regarded as absolutely free. We shall 

 further assume that p is the impulsive screw which has X as its instantaneous 

 screw, and that the relation of p to X is of the same nature. 



If, however, the four screws X, X , p, p possess the relations thus indi 

 cated, it is necessary that they satisfy the conditions already proved ( 281). 

 These are two-fold, and they are expressed by the following equations, as 

 already shown : 



cos (\p ) + P,*,. cos (\p f ) = 2r A v, 



cos (\p) cos 



p\ p\ 



We shall arbitrarily choose X and p, so as to satisfy the conditions 



A /P = 0, tnv = 0, 



and thus the second of the two equations is satisfied. These two equations 

 will give & as a function of &amp;lt;, and &amp;lt;/&amp;gt; as a function of 6. We can thus 

 eliminate & and &amp;lt; from the first of the two equations, and the result will 

 be a relation connecting 6 and $. This equation will exhibit the relation 

 between any instantaneous screw 6 on one cylindroid, and the corresponding 

 impulsive screw &amp;lt;/&amp;gt; on the other. 



It will be observed that when the two cylindroids are given, the required 

 equation is completely denned. The homographic relations of p and X is 

 thus completely determined by the geometrical relations of the two cylin 

 droids. 



