for resolving Transcendental Equations. 437 



any excentricity, which shall scarcely deviate perceptibly from the 

 true elliptic figure, so, in all these and similar cases, by substituting 

 elliptic for circular c.vcentric wheels, the arc u instead of being 

 directly related to the sines, cosines, tangents, &c, involved in the 

 equations may, without the slightest increase of mechanism, or 

 any additional difficulty in the process of solution, be replaced 

 by a transcendent of that form which depends on the rectifica- 

 tion of the ellipse. 



(21.) It is almost needless to mention that any mechanical con- 

 trivance which converts a uniform motion u into another not uni- 

 form, but varying according to any function <p (u) of the former, 

 affords either a solution of the equation <p (11) = A, or a tabulation 

 of the values of <p(u), just as we please. In the one case we have 

 only to arrest the motion at equal intervals of the scale on which 

 the graduation of A is engraved, and read off the graduation of 

 that on which u is represented. In the latter the movements 

 must be arrested at equal intervals of it, and the values of A read 

 off. Thus the tabulations of the direct and inverse functions pro- 

 ceed, pari passu. 



(22.) It is not my intention in this paper to enter at large 

 into the general question of the representation of analytical functions 

 by continued motion, though perhaps I may take a future oppor- 

 tunity of so doing. I will only here consider one other case, by 

 which, without any great complication of machinery, the principle 

 I have above adopted may be extended to equations containing 

 several transcendental relations, such as 



it + p . sin in 11 + q . sin 11 11 = A, 

 and others of the same nature. Suppose, instead of attaching our 

 excentric wheel to a point in the revolving arm JiC we attach it 

 to a second revolving arm, whose center of rotation occupies the 



8 k 2 



