342 A, Miikhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



apparently, therefore, the envelope is a pair of right lines passing through 

 the fixed point (a^, o), and real only if h is greater than unity, that is, 

 if the point S is outside the ellipse. But, looking to the geometry of the 

 figure, it is clear that the envelope must be the given point S, so that 

 the analytical solution furnishes, apparently, a whole line for the 

 envelope, while geometrically only one definite point on that line satisfies 

 the demand of the problem ; the discrepancy, however, is only apparent, 

 viz.^ the equation (167) may be written 



62 {x - a/i^)2 + a2 (^iTrp)2 ^2 ^ Q^ 

 so that this must be equivalent to 



x — ak\ 



2^=0 r 



which is, of course, the point in question. Such instances of degenerate 

 envelopes are by no means rare. 



§. 29. Properties of Confocals. 



I. Given a system of confocal ellipses, to find the locus of points 

 where the tangents cut off a constant area from the axes. 

 Any conic of the system is 



5+fi=i' (i^s) 



where, for the moment, 



A2 = a2 + X2, B2 = feHX2, c2 = A2-B2 = a2_?,2. 

 Take a point (^, >y) on this ellipse where the eccentric angle is <^ ; the 

 tangent is 



^ cos <^ + g sin </> = 1, 



and the intercepts made on the axes are 



A B 



cos <^' sin </>' 

 so that, if A2 be double the constant area in question, we have 



. ^ . ^h^ (169) 



sm <p cos 9 



Hence we get the system 



^2 = A2cos2<^=(a2 + X2) cos2</>, (170) 



',72 = B2sin2</> = (62 + X2) sin2<^, (171) 



and from (169) 



(a2 + X2)(62 + X2) = /i4sin2<^cos2</) (172) 



The elimination of X, <^ from these three equations will lead us to the 

 equation of the locus. For this purpose, observe that from (170) and 



(171), 



^2 v" = (»" + X^) (t* + >^^) sin* f oos2 # = ^* sin* ^ cos* ^, 



