FUNDAMENTAL THEORY OF COUPLES. 117 



III. To represent a couple by a straight line 



(i) In point of application. The position of a couple 

 with respect to a point is arbitrary, hence the straight 

 line may be drawn from any point. 



(ii) In direction. The straight line must be drawn* at 

 right angles to the plane of the couple, and in the direction 

 of translation of a right-handed screw which has the same 

 direction of rotation as the couple. 



(iii) In magnitude. The length of the straight line 

 must be proportional to the moment of the couple. 



This straight line is called the axis of the couple ; the 

 axis is therefore a straight line drawn from any point at 

 right angles to the plane of the couple, of length pro- 

 portional to the moment of the couple, and in the direction 

 of translation of a right-handed screw which turns in the 

 same direction as the couple. 



IV. To find the resultant of two couples represented 

 by their axes OG and OH (fig. 36) describe a parallelo- 

 gram on OG and OH as adjacent sides, then the diagonal 

 OK will represent the axis of the resultant couple. 



For, let the planes of the couples intersect in the line 

 ABj and take AB as the arm of each couple, and P and 

 Q as the forces. 



Find the resultants R of P and Q at A, and of Pand 

 Q at B; the two resultants are equal and opposite, and 

 constitute the resultant couple. 



OK is perpendicular to the plane of this couple, and 

 if OG = P.AB, OH^Q.AB, then will OK=R.AB; and 

 therefore OK is the axis of the resultant couple. 



Hence, to find the resultant of any number of forces, 

 find the resultant of their axes by the parallelogramic 

 law. 



The resultant, therefore, of a system of couples in the 

 same plane, or in parallel planes, is a couple of which the 

 moment is the algebraic sum of the moments of the separate 

 couples. 



