ENVELOPS. 361 



?n 



For all values of a comprised between -j-r r, - ^ and m, 



4/(m' + n') 



the circles do not ultimately intersect, and are not touched by 

 the envelop : for the value of y found from (2) and (3) is 



m 



in" 



which is impossible when a is greater than 



V \iu T /t ; 



Therefore in the enunciation of Art. 335 we do not assert 

 that the envelop touches each of the series of curves, but that 

 it touches each of the series of intersecting curves. The de- 

 monstration in that Article assumes that the equations (1) and 

 (2) lead to possible values of x and y; or in other words, that 

 one curve of the series ultimately intersects the adjacent curve. 



339. The method of Art. 334 may be extended to the case 

 in which there are n parameters connected by n 1 equations. 

 For example, suppose 



F(x> y, a, b, c) =0 (1) 



to be the equation to a curve, the parameters a, b, c, being 

 connected by the equations 



and that we require the locus of the ultimate intersections of 

 the curves obtained by giving to the parameters in (1) all 

 possible values consistent with (2). If from equations (2) we 

 find the values of b and c in terms of a and substitute them in 

 (1), we may then proceed as in Art. 334. If however the 

 solution of equations (2) be difficult we may proceed thus. 

 Regarding b and c in (1) as implicit functions of a, we have, 

 if we.differentiate with respect to a, and put the result equal 

 to zero as in Art. 334, 



da db da dc da 



(3) 



To find j- and -y- , we have by differentiating (2), 

 da da 



