The Elliptic Fauctions Defined. 67 



position; — merely a definite fraction of the circumference. We 

 may then continue bisecting the arcs of the circle of the argu- 

 ment, at the same time fixing on the other circle the points which 

 correspond to these points of bisection, until one of two things 

 must take place. Either one of the points of bisection falls on 

 the given point, in which case its corresponding point is un- 

 equivocally determined, or else two points of bisection will be 

 found, containing the given point between them, and such that 

 the arc of the circle of the amplitude, contained between their 

 corresponding points, is less than the assigned arc. But this 

 will be true, how small soever the assigned arc may have been. 

 There is then a limiting position on the circle of the amplitude, 

 corresponding to the given point on the circle of the argument. 



It may be remarked that to fix approximately the point cor- 

 responding to a given point, with sufficient accuracy for a 

 geometrical illustration, will not in general require many bisec- 

 tions, for the circles within the circle of the amplitude, inscribed 

 in the successive polygons, rapidly approach coincidence with 

 the latter circle, — as a result of which fact, a point within an arc 

 of this circle soon comes to divide the arc in sensibly the same 

 ratio in which the corresponding arc of the circle of the argu- 

 ment is divided by the corresponding point. 



Now let a point move on the circle of the argument with a 

 uniform velocity, and in a positive direction of rotation, starting 

 from the origin of arcs; and let a second point move on the 

 circle of the amplitude, starting from its origin, and moving so 

 that it always occupies the point corresponding to that occupied 

 at the same instant by the former moving point; whence it is 

 evident that its motion cannot be uniform, but will be more 

 rapid in the first part of the semi-circumference than in the 

 latter part. Then to any arc of the circle of the argument, be- 

 ginning at the origin, corresponds an arc of the circle of the 

 amplitude, beginning at its origin; viz.: they are the arcs 

 .traversed by the two points in the same interval of time. These 

 arcs are denoted by 2m and 2x respectively, and the halves of 



