282 Proceedings of the Royal Society of Edinburgh. [Sess. 
establishment of an integral erecting couple. The action of the device is 
illustrated further in fig. 15. As before, the pivoted system (fig. 12) is 
supposed tilted, with respect to the vertical, about the pivots JP2V2 ’ fhe 
erector being inclined towards the reader. As the erector rotates, each ball 
moves so that its centroid describes, relatively to the plate A, the path 
abcdef, where ahc and def are semicircles of radii and respec- 
tively. Consider the instant at which the balls occupy the positions 
shown. Clearly, if m is the mass of a ball, there is applied, at the instant. 
l^- 
Fig. 15. 
to the pivoted system a couple of moment mg — about an axis 
perpendicular to the line , and hence at the instant the gyroscope is 
precessing about an axis parallel to b-fi^, • fhe erector rotates, this couple 
rotates with it ; the moment of the couple remains constant, but the plane 
in which it acts moves round in azimuth at the speed of the erector. At 
the start of a half turn the couple is applied wholly about P2P2 j after a 
quarter turn it is applied wholly about p-^^p ^ ; and after the half turn it is 
again applied about P2P2 ^ but in the opposite direction. So long as b^ lies 
between a and b, and b^ between d and e, the applied couple has components 
about each of the axes and P2P2 • Call these Cj and . At the start of 
