248 



" In a note communicated some time since to the Academy, 

 I extended to the system of any hyperbola and its conjugate 

 a property of the equilateral hyperbola, and the lemniscate de- 

 rived from it, given by Mr. Talbot, in the fourteenth volume 

 of M. Gergonne's Annales de Mathematiques. The result 

 at which 1 then arrived I have since found to be a very parti- 

 cular case of a curious and general theorem, which may be 

 enunciated as follows : 



^^ Being given a hyperbola^ the equation of which is 



where a is supposed greater than b, let the curve be described, 

 which is the locus of the feet of perpendiculars dropped from 

 the centre upon its tangents : from this new curve let another 

 be derived, and so on, by repeating continually the above-men- 

 tioned construction, and let Sn denote the perimeter of the «'* 

 curve of the series. Also, let 2„ Je the perimeter of the n^ 

 curve, obtained by a similar inode of generation from the con- 

 jugate hyperbola, 



x^ y^ _ 



Then, any combination of these perimeters, such as 



will be expressible by elliptic functions of the first two kinds in 

 the following manner: 



Sn S2U1 + S2i S2.%, = 7r(a + /3F(A, (f) + 7E(A, ^)), 

 where 



A2= — — , COS^ = — , 



a^ + b^ a- 



and where a, /3, y, are algebraic functions of a and b. 



" The foregoing equation holds for the case ofi = 0, by sup- 

 posing that So {or So) expresses four times the difference be- 

 tween the infinite hyperbolic arc and its asymptote. 



*' It is evident that we may give a purely geometrical enun- 



