100 DYNAMICS OF A RIGID BODY. 



Whence we deduce Ah = -*% and since when p$ = o 



4 

 the screw must reduce to TJ, we find A = i . 



We, therefore, deduce the following theorem : If 

 rj!, .... *?6 be the co-ordinates of a screw of zero pitch, 

 then the co-ordinates of a screw ?, of pitch /^, upon 

 the same straight line as the screw j] are equal to the 

 six quantities : 



^_ ^ 



4-A d"n\ 4/6 dris 



in which 



R = rji z + . . . . + r} + 2rjiTj 2 COS (cui(i> 2 ) + 2i7 1 rj 3 COS fth6 + . . . = I. 



We may remark that the co-ordinates of a screw of 

 infinite pitch, parallel to ?, are proportional to : 



i dJR i dR 



We can also prove that -- is the cosine of the 



2 tflft 



angle between the screw r;, and the screw of reference Wl . 

 Let O be this angle, and let d be the shortest distance 

 between 17 and e^. Then we have ( 35) : 



and as this must be true whatever may be the value 

 of ps, it follows that : 



1 dR 



- = cos O. 



2 drji 



We also have the identity : 



J./'^.Y- -i^-o. 



' 



