The Elliptic Functions Defined. 59 



same principle then employed, would be similarly found on a 

 circle described from any other point of 1)Q as a centre, with 

 radius equal to the tangent to the outer circle from that point. 

 It is in fact a point common to all the circles that cut orthogon- 

 ally the system whose radical axis is DQ, and is fixed in posi- 

 tion as soon as the circle PB3I and the line DQ are given; or 

 conversely, this circle and the point G will determine the line 

 DQ. This circle, line, and point will be assumed as given in 

 position in subsequent constructions, and are to be regarded as 

 remaining unchanged in passing from one construction to 

 another. 



When in addition we have given a chord, as PM in Fig. 1, in 

 one of its positions, we are already able to construct the remain- 

 der of the triangle PMX, and the circle which, in the successive 

 positions of the triangle, is enveloped by 3IX. For we may first 

 construct the circle TT'W as directed in the construction 

 attached to Fig. 2; we may then locate T ou its circumference 

 by making PT = PT ; the line P7^ produced determines lYand 

 accordingly MN; and finally a second application of the con- 

 struction of Fig. 2 gives the circle to which MN must remain 

 tangent. As we have to attend only to circles interior to the 

 given circle, no ambiguity is encountered at any stage of the 

 process. 



If on the other hand the chord MN is given, and it is 

 required to complete the triangle by drawing PM and PN, we 

 may first construct, as above, the circle tangent to MN, — in the 

 figure, V-^SV.. — and then draw a tangent to this circle at 

 either of the points T^, or V., where it meets the axis. If M^Ni 

 is such a tangent, and PiM^ a line joining one of its extremities 

 to an extremity of the diameter P1P2, we can construct that circle 

 of the system which is tangent to Pjil/i ; and the symmetry of the 

 figure shows that the same circle will be tangent to P^Ni as well. 

 Accordingly, if tangent lines are drawn to the circle from M and 

 N, they must meet on the circumference of the given circle at a 

 point P, and the triangle will have been constructed as required. 



