385 



5. Let geodetic lines issuing from tiie same point upon a 

 line of curvature, and passing through the umbilics o, o', meet 

 the line of curvature again in the points p, p'. Then will the 

 locus of the point of concourse of the geodetic lines op', o'p, 

 be a line of curvature of the same species as the given one. 



The following note, by Mr. M. Roberts, on a theorem 

 relating to the Hyperbola, was also read : 



Let s denote the difference between the infinite arc and 

 the asymptote of the hyperbola, whose equation is 



^ ^ - 1 . 

 a- w 



and let s' be the length of the quadrant of the curve which is 



the locus of the feet of perpendiculars dropped from the centre 



upon its tangents ; also, let s, 2' denote the same things in 



reference to the conjugate hyperbola 



and we shall have 



ss' + 22' = ^7r|^ Vd^ — b'sl 



where we suppose a^b, and denote by s an arc of the first 

 hyperbola, measured from the vertex to the point whose coor- 

 dinates (x', y') are 



, _ a^ 

 00 - ^, 



y' = y^d' — b' 



\i a ■=■ h the hyperbola is equilateral ; the derived curve is the 

 common lemniscate, s = 2, s'lz 2'; and 



ss' = \-KC^, 

 a theorem proposed by Mr. W. H. Talbot, and proved by M. 

 Sturm, in vol. xiv. of Gergonne's Annales de Mathematiques, 

 page 17. 



Professor Harrison read the follovying paper on the ana- 

 tomy of the elephant : 



" Having had, within the last few weeks, an opportunity of 

 examining the body of an elephant which died in this city,, 

 2 K 2 



