300 Mr DA VIES on the Equations of Loci 



2. The inclination of an element of a curve to the meridian, is obtained 

 by the same considerations, viz. 



tenMSN -w= 



MN d6 sin , 



3. The area of a spherical curve is obtained by integrating 

 A = fil cos (j))dd + C 



Considering the elementary triangle whose sides are <p, $ + df, and in- 

 cluded angle dd, we have its area, by the usual rule for the spherical excess, 



. (b 4> + dd> . 

 an i sin r T Y Sln 



* * 



But, sind6 = d6; cos =1; 



Whence, cos + 



= cos ^ sin 



sin y ^ y = sin i- + i cos ^ d 6. 

 Making these substitutions, we find (1.) converted into 



sin 2 C sin + ^ d< ^ cos ~ 



+ 2 sin ^ (sin ^ +i cos-^ 



r / A \ 



(cos (p i sin 0<ty) + 2 sin | (sin 2 + i cos _^ d ^ " 



*v 



The infinitesimal addends of this being cancelled, the expression be- 



comes 



sin 2 d6 



cos ^> + 2 sin 2 -J- 



