The Elliptic Fu)ictio)is Defined. 



53 



impossible, and it follows that there is a region upon the line 

 joining the centres which is never crossed by the line whose 

 equation is the above equation (6); and moreover that the 

 limits of this region or segment are the points which we obtain 

 from (7) by giving to c the value R and — R successively; viz.,. 

 they are the points whose abscissas are 



m = 



i^[2r — (i? + a)-] 



and ))i" = 



i?[(i?— «V^— 2r] 



(8. 



{R + af ' ""^' '" {R — af 



These results are such as would be expected from a purely 



f'S I 



geometrical point of 

 view. If we construct 

 a diagram, Fig. 1, in 

 which two circles 

 P^PF.JL and TT'W 

 are drawn in any rela- 

 tive positions, and as- 

 signing to a point P 

 several different situa- 

 tions in succession 

 upon the circumference 

 of one of them, con- 

 struct, for each posi- 

 tion of P, the system 

 of chords as described 

 in the foregoing, it will be evident that corresponding to the 

 singly infinite succession of positions possible to P, there is a 

 singly infinite succession of positions of the line MX; and 

 accordingly it is to be expected that these successive lines 

 envelope some curve, which, in consequence of the general sym- 

 metry of the figure, would be symmetrical to the axis PoPi . 

 Moreover, when the point P is made to traverse the circum- 

 ference PiPM2L, a fixed point Q on the latter becomes twice 

 and only twice the extremity of one of the chords 3IN; that is, 

 once when the point M and again when the point N arrives at 

 Q. Hence it might be at least conjectured the that curve 



