for resolving Transcendental Equations. 429 



(6.) Let unity (1) represent the radius of the excentric wheel, 

 plus that of the thread, and e the length of the interval CB between 

 the centers of the index circle and excentric— or the excentricity 

 of the latter. Then if we draw AC parallel to the horizon, and 

 put u for the angle ACB, the perpendicular BG will = e . sin u. 

 Now, were the center of the excentric wheel preserved constantly 

 on the level of the line AC, and that wheel itself merely made 

 to revolve uniformly by the rotation of C about B instead of B 

 about C, the part of the thread which would be wound up on its 

 circumference by this rotation from the commencement of the motion 

 would be represented by lxu = u. This then would be the quan- 

 tity by which the vertical scale would, on that supposition, be 

 raised above its original position. But in the actual case, not only 

 is the thread so wound on the wheel, but the center B of the 

 wheel being raised above AC by e sin u, carries up with it the 

 wheel, thread, and scale, all by the same quantity. Therefore 

 the total elevation of the scale due to both causes acting at once 

 will be u+e.sinu. If then we put A for this elevation, there 

 will subsist between u and A the transcendental relation 



u + e . sin u = A, 



which is that of the problem of the excentric and mean anomalies 

 in the elliptic motion of a planet, the arcs being reckoned from 

 the aphelion. If we would reckon them from the perihelion, we 

 have only to wrap the thread the other way round the excentric 

 wheel, when the equation expressing the relation between A and ?/ 

 becomes 



u — e . sin u = A. 



(7.) It appears from what has been said, that the values of u 

 being read off on the index circle at H, those of A will be read 



3 I 2 



