186 



V. OSCILLATIONS AND CYCLIC MOTIONS. 



Let four equal masses, > be fastened to the ends of two mutu- 



ally perpendicular arms of negligible mass (Fig. 38), which are fastened 



rigidly where they cross, at 

 their middle points, to an axis 

 perpendicular to them both, 

 about which they turn. Let the 

 point of crossing of the three 

 arms be fixed while the system 

 can spin about the axis OP, 

 which can move in any manner. 

 We will suppose that during 

 the motion the axis OP makes 

 with the ^-axis a small angle 

 whose square can be neglected 

 in comparison with unity. Let 

 the position of the axis be 

 determined by the coordinates 

 !, 77, of the point in which it 

 intersects a plane perpendicular 

 to the ^-axis at unit distance 



Fig. as. from the origin. The squares 



and products of |, y, are con- 



sequently to be neglected. Let us further specify the position of the 

 system by the angle cp that the projection of the arm OA on the 

 XY- plane makes with the X-axis. Thus the three coordinates |, rj, cp 

 determine the position of the whole system. 



If the coordinates of the point A are x, y, 8, since it lies in a 

 plane whose normal passes through the point , ??, 1, we have 



174) g + lx 4- yy = 0. 



But since OA always makes a small angle with the XT- plane, the 

 projection of OA on this plane differs from it in length only by a 

 quantity of the second order, which we neglect. We therefore have 



Differentiating 174), 



dy = xdcp. 



so that we have 

 dx 2 + 



= xdl -f ydri -f (rjx ly) dcp, 

 + dz* = (Z 2 + rfx* + |y - 



