43-1 Sir J. F. W. IIerschel on a Machine 



(15.) Supposing all these adjustments well made, the workman- 

 ship good, and the graduations such as may be executed, there 

 seems no reason in the nature of the case why the mechanical 

 solution afforded by the apparatus we have above described should 

 not possess equal precision with any astronomical observation, and 

 therefore be available in many instances where extreme nicety of 

 computation is not required, or where several hypothetical ellipses 

 may require to be tried in the calculation of the orbit of a comet 

 or other celestial body. In the enquiries to which I have already 

 alluded I found in fact a very material saving of time and trouble 

 from the use of such an instrument, though constructed in the 

 rudest manner from materials casually at hand. 



(16.) The solution of the equation u + e . sin u — A includes 

 that of its derivative forms 



u + a . sin (u + b) = A, 

 and u + a . sin u + b . cos u = A, 



where however only sines and cosines of u are involved. If we 

 would introduce tangents, secants, &c. we must have recourse to 

 a modification of our mechanism. For instance, suppose the equa- 

 tion to be resolved were 



u + p . tan u = A, 



then, retaining the slider, excentric wheel, and horizontal axis C, as 

 in the contrivance already described, let the straight edge /./(fig. 2) 

 instead of being permanently fixed in a horizontal position, be made 

 to revolve on a center / vertically below C, with half the angular 

 velocity of the index hand, which may be done either by a toothed 

 wheel working into another of twice the diameter, or by catgut 

 wrapping tightly round cylinders in the same proportion, and let 

 the zero of both rotations be in the horizontal positions of CB and 



