Mr. W. Lupton on Spherical Geometry. 39 
7 
a | 
N 
Ma? 
L=+ ———,, 
a+a407)? etre ait i ls) 
M=+ ———, 
(1+a?+ 6?) 
a b 
as ee 32)? 
and the equation of a great circle is 
ax +by+1=0. 
7. The angle between two great circles is the angle between 
the planes by whose intersection with the sphere they are formed. 
Hence if the equations of the planes be 
LX +MY +NZ=0, 
UX +M'Y+N’Z=0, 
and the equations of the great circle be 
ax +by +1=0, 
we have ad'z + By +1=0, 
Base es. Ma a sy Ss RIG Sh ar) 
(1 +a? +62)? (1+ aa! + bb!) 
8. The angle between any two great circles is the distance 
between their poles. Hence if d be the distance between two 
points 2/y/ and ay" 
cos d= —— voustalie ve 
(1 + a!? 4 y!?)? (1 + all? + yll2)? 
9. The length of the perpendicular from a given point on a 
given great circle is the complement of the distance between the 
given point and the pole of the great circle. Hence if p be the 
perpendicular from the point 2’, y', on the great circle whose 
equation is aw + by+1=0, we have 
a. 1 + aa! + by! 
(1 4+a2+b%*(1 4+ai24 iz) ; 
In a future paper I shall investigate general expressions for the 
transformation of spherical coordinates, and I shall proceed to the 
discussion of properties of spherical curves of the second degree. 
ueen’s gis Galway, 
May 16, 1857. 
sin p 
