The Elliptic Fttnctions Defined. 55 



y be the variable eliminated, this condition appears at first in 

 the form 



{gnr + ^")" = ( ^' + m') {(j'-mr — 7»-p^ + )r) = 0; 



but is easily reduced to the simpler statement 



And if in this formula we substitute for // and ^3 the values 

 given in the equations (9), and for /, ui, and r, the coefficients of 

 equation (6), remembering that c" + s" = R-, we find that the 

 formula is identically satisfied; and thus it is shown that how- 

 ever the point c, s, may be situated on the circumference 

 X' + U' = H', the chord represented by (6) is tangent to the 

 circle whose position is defined by (9). The equation of this 

 ■circle, as written at the beginning of the present paragraph, 

 becomes, when r/ and p are replaced by their values, 



{R' — cry {x' + ?/-) + 8ar'E' x — R' (R' — crY 



+ 4:rR' (R' + a-) — 4.v'R- = 0. ^ 



To prove the second part of the theorem, we have only to 

 bear in mind that the equation of the radical axis of two circles 

 is obtained by subtracting the equation of one circle from that 

 of the other, having first multiplied either by such a" factor as 

 may be necessary to render the coefficients of x' and y- alike in 

 both. Combining in this way the equation x" -\- y~ — R- — Q 

 with the two equations (4) and (10) successively, we obtain in. 

 each case as the equation of the radical axis, 



2rw- + i?- + o- — ;-' = 0. (11 



Jacobi cites the foregoing theorem as due to J.-V. Pon- 

 celet, and occurring on page 326 of his celebrated work, " Tvaii6 

 des Propri6t6s Pi'ojccfives dcs Figures." In the edition of 

 1865 I find the theorem on page 315, in a form which may be 

 translated as follows: — 



" If an angle, being at the same time inscribed in one circle 

 and circumscribed about another, be made to move while sub- 

 ject continually to these same conditions, then the chord which 



