68 



Colorado College Studies. 



these arcs, viz., u and x, are named respectively the argument 

 and the amplitude. So that to find the amplitude correspond- 

 ing to any given argument, we have first to double the latter, 

 then to find the point on the circle of the amplitude correspond- 

 ing to the termination of the doubled arc, and finally to bisect 

 the arc included between the point so found and the origin. 



It is to be remarked that the argument, for which the cor- 

 responding amplitude is thus found, is given as an arc of a circle 



whose radius (if we make R unity) has been taken to be —^, 



hence its circumference is 4K. In other words, the argument 

 is supposed to be given, not in absolute value, but by its ratio 

 to a constant, K; just as an arc of the circle of unit-radius is 

 often most conveniently given by its ratio to -. When the point 

 moving on the circle of the argument has described the whole 

 circumference 4K, so that u = 2K, the point on the circle of 

 amplitude has also completed its circuit, and the relations of 

 position between these points for all greater arcs are the same 

 as for the arcs by which these exceed this respective circumfer- 

 ences, hence the amplitude of 'liiK + w exceeds by n- the am- 

 plitude of u. But as on the one hand it is totally unnecessary 

 to the foregoing construction to know the value of K, so con- 



V e r s e 1 y , the construction 

 affords no means of det er- 

 mining K. The considera- 

 tions by which a definite 

 value is assigned, though 

 external to the main purpose 

 of the present paper, will 

 be indicated in a concluding 

 paragraph. 



The fixed parts of the fig- 

 ure, whose relation of mag- 

 nitude is material to the. 

 construction, are only the circle of the amplitude and the radi- 



