80 Mr. C. V. Boys on Apparatus for the 



ring, the other being an inside ring, the barrels must not 

 be geared together. Fig. 2 is a plan on a larger scale, partly 

 in section, showing how such barrels might be supported. 

 Now, if this outside ring is placed horizontally within the 

 inside ring and touching it at one point, then revolution with- 

 out rotation of the inside ring will cause rotation without 

 revolution of the outside ring ; if, however, in consequence of 

 >he inclination of the tangent-wheel, the barrels of the inside 

 zing are caused to rotate, then such rotation will cause revolu 

 cion of the outside ring ; and this revolution will be a true 

 measure of the integral, as the outside ring and the tangent 

 wheel touch barrels of the inside ring at points having the 

 same radius. The astronomical convention with respect to 

 the terms revolution and rotation has been used. By revolu- 

 tion of the ring is meant a turning of the whole round a centre; 

 and by rotation a turning of barrels round their own axes. 



Instead of a disk, a sphere similarly mounted may be used 

 for a tangent-wheel, with the same result. Of course the 

 cylinder will be in contact with the sphere at a point on its 

 equator ; but if the support of this globe is varied in position, 

 so that the cylinder touches the sphere nearer the poles, then 

 the rate of rotation will depend not only on any former incli- 

 nation of the plane of the equator of the globe to the axis of 

 the cylinder, but will also be inversely proportional to the 

 cosine of the latitude of the point of contact. The latitude 

 should be brought back to its original value before the rota- 

 tion of the cylinder is measured. Fig. 3 shows the cylinder 

 in contact with tie sphere at a latitude A. It so happens that 

 the radius in the sphere at the point of contact is equal to 

 cos A, X the radius of the sphere ; but this is not the cause of 

 the introduction of that function, as the rotation of the cylinder 

 is independent of the radius of the tangent-wheel. The true 

 reason can be readily discovered by a simple geometrical con- 

 struction, which, from the length of this paper, I omit. How- 

 ever, a good illustration may be seen by taking a bicvcle and 

 causing it to lean over on its side; then a given twist of the 

 handles will be found to produce a greater deviation in the 

 direction of its motion than would be the case if the bicycle 

 were upright. The effect just described is most easily pro- 

 duced by mounting the cylinder on a rocking-frame, so that it 

 can roll round the ball. Though the axis marked A in the 

 figure remains vertical, yet the effect may be considered as 

 due to a leaning to one side of this axis. If, however, the 

 axis A is caused to lean forwards or backwards, then the rota- 

 tion of the cylinder, which is still proportional to the tangent 

 of any rotation about A, is also proportional to the sine of the 



