330 THE THEORY OF SCREWS. [306, 



ray in space, and assign to it an arbitrary pitch, the screw so formed may be 

 regarded as an impulsive screw, and the corresponding instantaneous screw 

 will not, in general, be defined. There is, however, a particular pitch for 

 each such screw, which will constitute it a member of the system of the 

 fifth order. It follows that any ray in space, when it receives the proper 

 pitch, will be such that an impulsive wrench thereon would set any one 

 of the singly infinite system of rigid bodies twisting about the same 

 screw a. 



307. The Geometrical Theory of Three Pairs of Screws. 



We can now show how, when three pairs of corresponding impulsive 

 screws and instantaneous screws are given, the instantaneous screw, corre 

 sponding to any impulsive screw, is geometrically constructed. 



The solution depends upon the following proposition, which I have set 

 down in its general form, though the application to be made of it is somewhat 

 specialized. 



Given any two independent systems of screws of the third order, Pand Q. 

 Let &) be any screw which does not belong either to P or to Q, then it is 

 possible to find in one way, but only in one, a screw 6, belonging to P, and a 

 screw &amp;lt;j&amp;gt;, belonging to Q, such that o&amp;gt;, 6 and (f&amp;gt; shall all lie on the same cylin- 

 droid. This is proved as follows. 



Draw the system of screws of the third order, P , which is reciprocal to P, 

 and the system Q , which is reciprocal to Q. The screws belonging to P , 

 and which are at the same time reciprocal to a&amp;gt;, constitute a group reciprocal 

 to four given screws. They, therefore, lie on a cylindroid which we call P . 

 In like manner, since Q is a system of the third order, the screws that can be 

 selected from it, so as to be at the same time reciprocal to &&amp;gt;, will also form a 

 cylindroid which we call Q . 



It is generally a possible and determinate problem to find, among the 

 screws of a system of the third order, one screw which shall be reciprocal 

 to every screw, on an arbitrary cylindroid. For, take three screws from the 

 system reciprocal to the given system of the third order, and two screws on 

 the given cylindroid. As a single screw can be found reciprocal to any five 

 screws, the screw reciprocal to the five just mentioned will be the screw now 

 desired. 



We apply this principle to obtain the screw 6, in the system P, which is 

 reciprocal to the cylindroid Q . In like manner, we find the screw &amp;lt;/&amp;gt;, in the 

 system Q, which is reciprocal to the cylindroid P . 



From the construction it is evident that the three screws 0, &amp;lt;f&amp;gt;, and &amp;lt;w are 

 all reciprocal to the two cylindroids P and Q . This is, of course, equivalent 

 to the statement that 0, &amp;lt;j&amp;gt;, a&amp;gt; are all reciprocal to the screws of a system of 



