129 



x' antl y, supposing x and y to be constant, we slioulil find 

 another equation, which, taken along with that of the normal 

 arc, would furnish the values of x and y, the coordinates of 

 the point in which two consecutive normal arcs intersect : 

 and thus, as in plane geometry, we find the evolute of a 

 spherical curve. 



Let 2-y be the diametral arc of the circle of the sphere 

 which osculates a spherical curve at the point oc', y', Mr. 

 Graves finds that 



^ r</a/2 + dy'^ + {x'dy' - y'dx'f]^ 

 tan y =. ± ■ ■ 



(1 + x''^ + y"')lidx'dhj' - dy'd'x')' 



For the rectification and quadrature of a spherical curve 

 given by an equation between rectangular coordinates, the 

 following formulae are to be employed : — 



yJ dx'- + dy"^ + {x'di/ - y'dx'f 

 ds =: 



and 



rf{area) = 



1 + X'-' + y'' 

 ydx 



(1 + X-) y/ 1 + x' + y- 



In the preceding equations the radius of the sphere has 

 been supposed rr 1. 



The method of coordinates here employed by Mr. Graves 

 is entirely distinct from that which is developed by Mr. 

 Davies in a paper in the 12th Vol. of the Ti'ansactions of the 

 Royal Society of Edinburgh. Mr. Graves apprehends, how- 

 ever, that he has been anticipated in the choice of these 

 coordinates by M. Gudermann of Cleves, who is the author 

 of an " Outline of Analytic Spherics," which Mr. Graves 

 has been unable to procure. 



The President communicated a new demonstration of 

 l;ourier's theorem. 



