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



it in its horizontal position (in which also CB must be adjusted 

 to be accurately horizontal,) and the parts of this scale must be 

 double of the corresponding- parts of A, that is, double degrees of u. 

 Each part therefore must be one 180th part of the circumference of 

 the circle DEF, as measured by the unwinding of the thread in 

 the mode explained in (Art. 8.) 



(19.) If the equation proposed were the rather more general one 



u + p . tan u + q . sec u = A, 



we might employ the same construction slightly modified by making 

 the straight edge IJ revolve attached to an arm at right angles 

 to the axis of the lower wheel, so that the motion of rotation of IJ 

 shall as before be uniform, and half as swift as that of CB, but 

 its direction not passing through the center Q of its revolutions. 

 All other things remaining as in (Art. 16.), let the perpendicular 

 QI =f then will the value of MJ be as before, 



MJ = 2 u + e sin 2 u + (CL - CG - GK.) . tan u, 

 but in this case we have 



CL =f. cosec u + b . cotan u, 

 so that the value of MJ becomes 

 MJ= 2 ii + e . sin 2 u + \f. cosec u + b . cotan u + e . cos 2 u - 1 ] . tan u, 

 =(2 u + b) + (e-l) . tan u +f. sec u, 

 that is, putting MJ =2A+b; e = 2p + 1 ; /= 2q, 

 A = u + p . tan u + q . sec u. 



(20.) It is needless to enlarge on the adjustment of these con- 

 trivances, which are merely introduced as specimens of the varia- 

 tions which a trifling change in the construction of our mechanism 

 is capable of making in the form of the equations resolved. I 

 will only observe here, that as ovals can now be turned of almost 



