traced upon the Surface of the Sphere. 275 



III. 



We may from this and a known theorem in spherics, express the area 

 of a spherical triangle, in terms of the co-ordinates of the three angles of 

 that triangle. 



* 



For, let a, (3,, a tl /?, and a tll fi llt be the three pairs of co-ordinates, then 

 we have as above, 



cos 8, = cos a u cos a tll + sin a u sin a M cos /? (3 M } 



cos 5,, = cos a, cos a //; + sin a, sina^cos (3, ^ tll \ 



cos 8,,, = cos a., cos a,, + sin a, sin a /; cos /3, j9 a J 



But, combining the expressions of Du GUA and CAGNOLI for the sphe- 

 rical excess, we have 



1 + COS 8, + COS 8 a + COS 8,,, 



COS -- - ' " ~ ""'" j *. I **. I 



2 ^ 2 (1 + cos d t ) (1 + cos ) (1 + cos ,) 



which gives, by means of (1.), the following symmetrical formula for the 

 spherical excess, in terms of the co-ordinates of the angles of the triangle, 



s l + rosg / ros // +cosa / )sg ;< +cos, ) cos /</ +sin / sin, ) cos/3 ) -/3 // +sina^ 

 os- . ' _(6.) 



V ^(l+cos^cosa^+sina.sina^cos^-^Jfl+cos^cosa^+sina.sina^cos-^l+cosa^cosa^+sina^sina^cos/S 



By means of a series of subsidiary theorems, I have been able to give this 

 a form adapted to logarithms ; but as the purpose of this paper has no refer- 

 ence to facility of calculation, and as, moreover, the process is long, and con- 

 nected with inquiries which I may bring separately under the consideration 

 of the Royal Society, it will be unnecessary to enter upon it here. In its 

 present form, it will not, I think, be without interest to mathematicians. 



IV. 



To find the equation of a great circle passing through two given 

 points on the sphere. 



Let K, \ be the unknown co-ordinates of the centre of the circle whose 

 equation we seek, and let a, (3, and a,, (3,, be the co-ordinates of the two given 



