GEOMETRY OF MOTION. 



[72. 



The sense of the vector is determined as follows. In the case 

 of infinitesimal rotations it appears from Arts. 67 and 64 that a 



couple of the type A, Fig. 24, 

 produces a translation upwards 

 from the plane of the figure, 

 i.e. towards the reader ; while a 

 couple of the type B produces 

 a downward translation, away 

 from the reader. Regarding the couples as rigid figures, their 

 rotors as forces, and the middle point of their arms as fixed, 

 the type A tends to produce rotation in the counter-clockwise, 

 positive, sense ; the type B in the negative sense. The former 

 is therefore regarded as positive, and its vector is drawn from 

 its plane towards the reader. 



Fig. 24. 



72. Let us now return to our infinitesimal displacements. 



An infinitesimal translation ds can be combined with an infini- 

 tesimal rotation d# about an axis 1 at right angles to ds (Fig. 25). 

 To find the resultant single rotation we have only to replace 

 the translation ds by an equivalent couple 

 whose angle of rotation we select equal to 

 that of the given rotation ; that is, we put 

 ds = L,L-dd, whence 



LI - ds 



L -' 



d6 



-d8 



The plane of the couple, being perpen- 

 dicular to ds, can be taken so as to contain 

 the axis / of the given rotation dO ; and in 

 this plane the couple can be so placed that p ig 25. 



one of its sides (see Fig. 25) falls into this 

 axis /. Selecting the proper side of the couple, we shall have on 

 /two equal and opposite rotations d6,d0, which destroy each 

 other, leaving only the rotation d6, about an axis at the distance 

 from /. 



