Ball — Contrihutiona to the Theory of Screws. 23 



vector-screw a, while at the same time tlie body is acted upon by a wrench of 

 intensity j3" on another vector-screw /3. 



We denote by Q the right-handed angle between n and /3. We take P a 

 point on a, and Q a point on j3, such that PQ is at right angles both to u and 

 /3, and of course PQ = d is the shortest distance between a and /3. 



The wrench on /3 is composed 



(1) of a force of j3" units acting on Q, and in the direction of the vector 



on /3 ; and 



(2) of a couple of moment ^"-p^ right-handed or left-handed about the 



vector on /j, according as jS''^^^ is positive or negative. 



The couple may be represented by the two forces (5"pfid'^ separated by the 

 distance d. One of these forces acts at P and the other at,Q; and they are 

 normal both to (5 and PQ. We sacrifice no generality by making both /3" 

 and pp positive, so that the couple shall tend to a right-handed rotation 

 about |3. 



The original wrench on /3 is thus replaced by three equivalent forces, viz. : — 



ft" acting on Q, 

 ft"ppd'^ acting on Q, 

 ft"p^dr^ acting on P. 



As all the forces forming the wrench are thus brought to bear either at P 

 or at Q, it follows that in the determination of the virtual moment we are only 

 concerned with the displacements of the two points P and Q. 



The twist about the vector- screw a gives to P a displacement a'p^ in the 

 direction of the vector on a. The same twist produces two displacements in 

 Q, to wit, a'pay in the direction of the vector on a, and ad, which lies in the 

 normal to a and PQ, and tends in the direction of a right-handed rotation 

 about a. 



There is thus one virtual moment at P, but there are four at Q ; for each of 

 the two displacements of Q will have a virtual moment with each of the two 

 forces at Q. The algebraic sum of these five quantities expresses the vii-tual 

 moment of the original twist and the original wrench. 



Fig. 2 shows the vector-screw a projected g 



down on the plane of the paper, which is 



supposed to contain ft, and to which QP 



is normal. The actual situation of a is 



above the plane of the paper; and this is '^ 



suggested in fig. 2 by making « longer than 



ft. With this agreement it is obvious from the definition that and not 



"iGO"- Q is the ririlit-handcd angle between the two screws a and ft. 



B. I. A. PROC, VOL. XXVin., SECT. A. [4] 



