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 Annates de Mathematiques. The result 

 at which I 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 w"' 

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

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

 jugate hyperbola, 



a2~62"~ 

 Then, any combination of these perimeters, such as 



S'ii S2i'+l + 22i S21+I, 



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

 the following manner: 



S2i Sii.i + S2i ^2i.i = 7r[a + (5F{k, <j,) + 7E(*, 0)}, 



where 



^ 2 , A2 » COS0=— , 



a^ + 0^ ^ a^ 



and where a, j3, y, are algebraic Junctions of a and b. 



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

 pas hig 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- 



