The Elliptic Functions Defined. 



.57 



f'f 2 



PM in T. This is the point of 

 contact. From T draw Tc per- 

 pendicular to PM, and meeting 

 CD in c. The circle with center 

 c and radius c T is the circle re- 

 quired. For it touches PM, 

 while the tangents QR and QT 

 are equal, so that QD is, as re- 

 quired, the locus of equal tan- 

 gents of the two circles. 



There are two solutions. For 

 the arc RTO, extended, will meet 

 PQ produced, if the produced 

 part be made equal to TQ. We may state this as follows: 



Of all the circles having a given radical axis in common, two 

 and only two are tangent to any given line, and their points of 

 contact lie on opposite sides of the radical axis. 



In all of the foregoing the relative positions of the circles 

 considered, or, in the analytical part, the relative values of i?,. 

 r and a, have been subject to no limitation. We have now, 

 however, to impose a restriction in this respect. 



Theorem. — When two chords of a circle, which meet on its 

 circumference, are tangent to a second circle which lies wholly 

 within the former, the third circle, which is touched by the line 

 joining the extremities of these chords, from whatsoever point 

 of the circumference they are drawn, lies also wholly within the 

 first circle. 



If this joining line be MN of Fig. 1, (supposed indefinitely 

 produced) it is, in any given position, as we have just seen, 

 tangent to hvo of the circles which have a common radical axis 

 with the two given circles. If the point P be moved, and the 

 line MN move in consequence, it cannot of course remain 

 tangent to both; and the one to which it does remain tangent 

 has been already found; being uniquely determined, from among 

 all the circles having the before-mentioned radical axis, by either 

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