7i ] ROTATIONS. 37 



parallel rotors is explained by the example of infinitesimal rota- 

 tions. The student acquainted with elementary physics will 

 recognize in this rule the so-called principle of the lever which 

 is based on the composition of parallel forces. 



70. Two rotors of equal length and opposite sense situated on 

 parallel lines (Fig. 23) are said to form a couple. The two rotors 

 P y P are called the sides, their perpen- 



dicular distance / the arm, and the 

 product Pp the moment of the couple. 

 It has been proved in Art. 67 that a 

 couple of infinitesimal rotations pro- 

 duces an infinitesimal translation. In 



general, a rotor couple is equivalent to a 



, n i 

 vector, as we shall see later. 



71. The converse proposition of Art. 67, viz. that an infini- 

 tesimal translation can always be replaced by a couple of 

 infinitesimal rotations, requires a little further consideration. 



Suppose we wish to replace the translation ds by a couple. 

 According to Art. 67, the axes / lf / 2 of the two rotations must 

 be at right angles to ds ; the distance L^L^ of the axes and the 

 angle of rotation ad are only subject to the condition that their 

 product should equal ds, i.e. 



There is, therefore, an infinite number of couples equivalent to 

 ds, all having the same moment L^L^ ds and all lying in a plane 

 perpendicular to ds. 



It thus appears that the characteristics of a couple are its 

 moment and the aspect of its plane ; in other words, a couple 

 (P, p) is equivalent to any couple (P 1 ', /') provided (a) that they 

 lie in parallel planes or in the same plane, and (b) that their 

 moments are equal, i.e. />/ = /"./'. This allows us to repre- 

 sent a rotor couple (P, p) by a vector perpendicular to the plane of 

 the couple and equal in magnitude to its moment Pp. 



