24 



Proceedings of the Royal Irish Academy. 



Fig. 3 is in the plane drawn through P and normal to PQ. Thus fig. 3 

 represents a plane above the plane of fig. 2. 



Fig. 4 is in the plane of fig. 2 ; i.e., it is the plane drawn through Q and 

 normal to QP. 



In figs. 3 and 4 we represent the three forces into which the wrench has 

 been analyzed, and also the one displacement of P and the two displacements 

 of Q. The displacements are shown by the dotted lines and the forces by 

 continuous lines. 



As the body is translated through the distance a f^ ; in tlie direction of o 

 this displacement must be assigned both to P in fig. 3, and to Q in fig. 4. 



The rotation a about a is without eflect on P, and therefore does not 

 appear in fig. 3. But this rotation displaces Q, as sliown in fig. 4, through 

 the distance ad. The dii'ection of this displacement in fig. 4 is determined 

 by remembering that the rotation is right-handed about a, and that a is 

 above the plane of the paper in fig. 4. 





..y.- 



,■>" 



<*6 



'''^■^ 



Fir,. 



Fio. 4. 



The directions given to the forces fi"p»'l'^ in figs. 3 and 4 are such as to 

 make the couple which they form right-handed with regard to /3. 



The virtual moment at P is accordingly expressed by the single term 



- a' ^' paPfid-^ sind. 

 The virtual moment at Q, fig. 4, is the sum of 4 terms. 

 The virtual moment of /3" and «'/>. is + n'/3"/>.cos B, 

 /3" ,. oV „ -a'/i'VsinO, 



„ /3'>W' " n'y. .. + u'ii"paPffi'' sin e, 



Assembling the five terms, which collectively form the virtual moment 

 of tlie wrench on /3 and the twist about a, it is seen that the virtual 



