ALONG THE CIRCUMFERENCE OF A CIRCLE. 457 



are enabled to compute the time at which a heavy body moving along the cir- 

 cumference of a circle will reach a given point in it, when the velocity is 

 insufficient to carry the mass over the zenith point. 



The other case, when the body passes the zenith point, is resolved by con- 

 sidering the angles at a, A, C, &c. Thus, if we describe a circle having its 

 diameter inversely proportional to the square of Ac, and imagine a body to fall 

 from a height greater than that diameter in the ratio of dA : Ac, and to be pro- 

 jected along the circumference with the velocity so acquired, its motion would 

 be represented by the variation of the angle Acd. 



But this same investigation enables us also to resolve the converse problem : 

 " to find the position of the body at any given instant," for since the time of 

 describing any portion of the arc NZ, fig. 1 1 is proportional to a logarithmic 

 tangent, we can easily compute the log tangent, and thence the arc corre- 

 sponding to a given time ; and having thus obtained the angle IHG of figure 

 10, we can deduce, by the operations of ordinary trigonometry, all the other 

 angles, and thus the positions of the various bodies at the proposed instant on 

 all the other circumferences can be found. 



In retracing backwards the series of trigons, beginning with GIH, we must 

 make IGF double of IGH, GED double of GEF, and so on ; hence we must 

 soon obtain an obtuse angle, the double of which would be a reverse angle, 

 and it would seem as if our process failed. But in reality this reverse angle 

 merely shows that the moving body has overpassed the zenith point, and has 

 begun to descend along the other circumference, and thus our construction 

 turns out to be absolutely general. 



When one of the angles, as ECD of figure 10, becomes right, its double 

 ECD becomes half a revolution, and so CAB and CBA become zeroes. 

 Wherefore when the motion represented by the trigon CED has performed half 

 of its period, the system typified by CAB has made a whole one. Thus at each 

 step downwards along the series the periodic time is doubled ; and if we wish 

 to keep the same periodic time throughout, we must halve the dimensions of 

 each successive trigon— an operation which brings us exactly to the conclusions 

 obtained in the original paper. 



VOL. XXVI. PART II. 6 C 



