438 Sir J. F. W. Herschel on a Machine 



point which that of the excentric itself occupied in the original 

 construction, and let this second arm have a velocity of rotation 

 in a constant ratio to that of the first, a condition attainable by 

 contrivances to be presently considered. In that case our construc- 

 tion will be as in fig. (4), respecting which figure we will establish 

 the following notation. 



CB = e; BB = e; BE' = 1 ; angle ACB = au; ~DBB' = pu; 

 MK = x, and DEM = c, 

 when we have 



MK = x = D'EK - DEM = HE' + BG - DEM, 



= ( a + fi)u + BT+ BV-c, 

 = (a + /3) u - c + e . sin a u + e . sin (o + /3) u, 

 assume therefore a = m; a + /3 = w; e = np; e' = nq; x = nA - c, 

 and the relation between u and A will be that proposed, viz 



u + p . sin mu + q . sin nu = A. 



This equation, it will be observed, can always be so prepared as 

 to make m and n integers, or, if we prefer it, fractions, whose de- 

 nominators are integers. The form however which will require 

 the least apparatus of wheels, and into which it is easily trans- 

 formed, is the following : 



A = u + p . sin u + q . sin n u, 



in which n is less than unity; for in this state of the equation 

 the first mover may be applied at once to the axis C, which, as 

 in the former construction, may carry an index arm reading oft* u 

 in degrees on a circle concentric with it. The whole difficulty 

 then is reduced to the solution of a mechanical problem. To com- 

 municate to the arm BB revolving on a center B, attached to an- 

 other revolving arm CB, a rotation having a given ratio of velocity 



