318J THE GEOMETRICAL THEORY. 345 



the fulfilment of two identical conditions among their coordinates. As 

 there are three pairs of correspondents, we see at once that there are six 

 equations to be fulfilled. These are exactly the number required for the 

 determination of rj, , , in the system P. 



To the same conclusion we might have been conducted by a different 

 line of reasoning. It is known that, for the complete specification of a 

 system of the third order, nine co-ordinates are necessary ( 75). This is 

 the same number as is required for the specification of a rigid body. If, 

 therefore, we are given that P, a system of the third order, is the collection 

 of impulsive screws, corresponding to the instantaneous screws in the 

 system A, we are given nine data towards the determination of a rigid 

 body, for which A and P would possess the desired relation. It therefore 

 follows that we have nine equations, while the rigid body involves nine 

 unknowns. Thus we are led to expect that the number of bodies, for which 

 the arrangement would be possible, is finite. When such a body is de 

 termined, then of course the correlation of the screws on the two systems 

 is immediately accomplished. It thus appears that the general problem 

 of correlating the screws on any two given systems of the third order, 

 A and P, into possible pairs of impulsive screws and instantaneous screws, 

 ought not to admit of more than a finite number of solutions. 



We are now to prove that this finite number of solutions cannot be 

 different from unity. 



For, let us suppose that a screw X, belonging to A, had two screws 

 6 and &amp;lt;f&amp;gt;, as possible correspondents in P. This could, of course, in no case 

 be possible with the same rigid body. We shall show that it could not 

 even be possible with two rigid bodies, M { and M 2 . For, if two bodies could 

 do what is suggested, then it can be shown that there are a singly infinite 

 number of possible bodies, each of which would afford a different solution of 

 the problem. 



We could design a rigid body in the following manner : 



Increase the density of every element of Mj in the ratio of p^ : 1, and 

 call the new mass M. 



Increase the density of every element of M 2 in the ratio of p 2 : 1, and 

 call the new mass M 2 . 



Let the two bodies, so altered, be conceived bound rigidly together by 

 bonds which are regarded as imponderable. 



Let ty be any screw lying on the cylindroid (6, &amp;lt;j&amp;gt;). Then the impulsive 

 wrench of intensity, ty &quot; on i/r, may be decomposed into components 



, sin (^r-&amp;lt;J&amp;gt;) , , sin (6 - ^) 



V /d ,( on 9, and iK -^ ^ *g on &amp;lt;f&amp;gt;. 

 sin (6 - &amp;lt;/&amp;gt;) sm(0 - &amp;lt;) 



