129 



x' and y , supposing x and y to be constant, we should 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^ be the diametral arc of the circle of the sphere 

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

 Graves finds that 



_ -J- [d^'' + d^J'^ + («%' - y'dx'f]l 

 tan y ^ nz 



(1 + X''' + y"')i{dx'<Py' - dy'ifx') 



For the rectification and quadrature of a spherical curve 

 given by an equation between rectangular coordinates, the 

 following formulae are to be employed : — 



and 



_ ^/ dx'-' 4- dy'^ + {x'dt/ - y'dx'f 

 d (area) = ^ 



(1 + a;2) V 1 + x^ + y- 



In the preceding equations the radius of the sphere has 

 been supposed = 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 Transactions of ihe 

 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 

 !^ourier's theorem. 



