The Elliptic Functions Defined. 51 



the desired quantities; and the two remaining equations may- 

 then be used to verify these results. We find 



Ac — Be + 2Hs —AS+ Bs + 2Hc 



I = 



■> I •! J '" — T, ; :; 



C + 8" C" + S" 



^-1 — X), k = — 



If now the above values of /, m, and n be substituted in th& 



equation 



Ix + mil -\- n — 



and the latter cleared of fractions by multiplying through by 

 c" + s'-, we have the sought equation of the third side of the 

 inscribed triangle, viz., 



(2JJs + Ac — Be) jc + (2H"c — .4s + Bs) u = {A -^ B){& + s"). (3 



Problem. — If two chords of a given circle meet on its cir- 

 cumference, and are both of them tangents to a second given 

 circle, to write the equation of the line which joins the extremi- 

 ties of these chords. 



This is the same as the foregoing problem, with the added 

 condition that the two lines which meet in the point c, 8, are to 

 be tangent to a second given circle. Let the radius of this 

 second circle be r, and its centre situated at a distance a from 

 that of the former; but, for subsequent convenience, let the sign 

 of a be taken opposite to that of the abscissa of the centre, 

 which we assume to be situated on the axis of X. Then the 

 equation of this second circle is 



.r + if -\- 2a.» + rt" — ?'^ = 0; (4 



and that of the pair of tangents to it from the point c, s is 



(r^ — s^) £c' + 2 {as + cs) xy + {r- — a^ — 2rtc — c") if 



— 2 (rts^ + cr^) x-\-2 {a^s — ■/•^s + acs) y (5- 



+ (c-r+r-s- — «V)=0. 



(I assume the latter equation as known, since it is given in the 

 text-books, but I may remark that a handy method of obtaining 

 it is identical in principle with that employed in the former 



