for resolving Transcendental Equations. 433 



the horizontal situation of BC, or of the cross-piece level, the 

 line BC being imaginary and intangible. Good workmanship will 

 of course ensure a very near approach to parallelism between this 

 line and the two sides of the cross-piece ; but if an error be still 

 supposed to exist it may be detected by the following process : 

 make e = 1, or set the slider to 100, and then beginning at 0° of 

 the index read off the value of A, corresponding to equal small 

 increments of u (for example from degree to degree) round a com- 

 plete semicircle to 180°. Then, since we have 



d A (PA 



-^ = 1 + cos u, and ^- = - sin u, 



a comparison of differences and second differences will readily 

 enable us to perceive whether any appreciable deviation from 

 the true position exists. It is only by the differences of read- 

 ings that an error in the zero of u can be separated from one in 

 that of A, (in which is included that of the index reading at H), 

 as the form of the equation u + e . sin u = A will easily make 

 evident. 



(14.) However truly the cylindrical form of the excentric wheel 

 is attained, if the axis of the cylinder be not parallel to that of 

 the index circle, the thread will wind off an ellipse in place of a 

 circular arc, of which equal portions will not correspond to equal 

 angles of rotation. This then affords a means of detecting and 

 rectifying such want of parallelism in the axes in question ; but 

 as its effect would be confounded with that arising from the error 

 in the origin of u just mentioned in the last article, it will be pre- 

 ferable to examine through a whole revolution of the index axis 

 (which should be perfectly horizontal) whether the thread main- 

 tains precisely the same distance from either edge of the excentric 

 wheel as it wraps itself round it. 



