352 ON THE MOTION OF 



motion now considered introduces among the variations of the 

 six elements of position*. 



The same results may be obtained immediately from the 

 equation bv = o (which is the fundamental equation of this 

 kind of motion) taken in connection with the value of bv given 

 above. It is evident that we have also in this case 



bx, = bx n by, — by,, bz, = bz,. 



If there be a second surface of support upon which also the body 

 is to roll without sliding, we shall have three other equations 

 exactly similar to the above. If we denote by (5a,, b(3,, by, 

 the variations in the direction of the body-axes of a fixed point 

 in the body whose body-coordinates referred to O, are a, % y, 

 we shall obtain from equations (12) 



da, = d^+ydQ — pdR, 

 b(3, = b^-habQ — ybP, 

 by, = bt,-)-(3bQ — abQ. 



From which if we subtract (25) we have 



da. = ( Y -z)bQ-(P-y)bR, 

 b{3> = {a — x)bR—(y—z)bP, 

 by, = (p—y)dP — {a—x)bQ. 



For the points of the body which are momentarily at rest, both 

 sides of these equations become equal to nought, and we obtain 



dP — <1Q — dR ' 



the equation of a straight line passing through the point of 

 contact and parallel to the axis of instantaneous rotation. 



* Since this communication was handed to the Librarian to be read be- 

 fore the Society, Mr E. Nulty has shown me the above three formulas, derived 

 (in the solution of a problem that had recently occupied his attention) from the 

 ingenious consideration that in perfect rolling the motion of the physical point of 

 contact in the direction of the body-axes is equal and opposite to the motion of the 

 point in which these axes arc supposed to have their origin. 



