Ball — Contributions to the Theory of Scretvs. 



29 



But this must be true for all values of i\ whence 

 cos 6 = and d = Q, 

 and accordingly ») must also intersect 7 at right angles. 



We shall now investigate generally the conditions which must be satisfied 

 if a body having received three twists of amplitudes a, /3', 7' about tluee 

 vector-screws a, j3, y respectively is to be restored by the third twist to the 

 position it occupied before the first. 



It has just been shown that the conditions require o, /3, 7 to be 

 intersected at right angles by a common axis which we shall suppose to be 

 normal to the plane of the paper at 0, fig. 6. 



^ \ 



X 



I 



' 1 1 



' I 1 

 /' ' ' 



Fio. 6. 



We now take a screw jj which is subject to no other limitation than that 

 it shall also intersect the same axis at right angles. 



In drawing the figure I have made 



Or, > Oa > 0(5 > Oy, 

 as has been already indicated. This is the conventional device for indicating 

 that Ot] is higher above the plane of the paper than Oa, that Oa is liigher than 

 Oj3, and that 0/3 is higher than Oy. 



By 0^ we may understand the angle measured from XII on a watch-dial 

 in the direction of increasing figures to Oij which is the projection of i; on the 

 dial. Similar meanings attach to 6^, 0^, 6y. 



As I) is above a, the right-handed angle between rj and « is 



IjOa = 0„ - By. 



The respective distances of »» and a above the plane of the paper' are «, and :r„, 

 and as the former of these is the greater, the shortest distance between 7; and « 



