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PROCEEDINGS OF THE AMERICAN ACADEMY 



2. It is evident that, in this system of co-ordinates, a straight line 

 in the phme will be projected into an arc whose equation will be of 

 the first degree in spherical co-ordinates, and that in general a locus of 

 the «th degree in the plane will be represented by one of the nth 

 degree in spherical co-ordinates. The equation of a great circle may 



then be written 



Ax-\-By=\. (a) 



Let p be the pole of this circle; 

 and let the co-ordinates of p be 

 tan Oa' and tan ob'. But oa' = 

 OA — 90°, OB' = OB — 90«*. 

 y^ Hence, the co-ordinates of the pole 

 are — A and — B ; that is, when 

 the equation of a circle is written 

 in the form (a), the co-ordinates 

 of its pole are the negatives of the 

 coefficients of x and y. This prin- 

 ciple is of great utility in finding distances and equations on the sphere. 

 The equation of a circle can also be written y = mx -\- n; and from 

 this it can be deduced, as in plane co-ordinates, that the equation of an 

 arc through a jjoint x'y' is 



y — y' = m(x — x% 



and of that through two points is 



y — y' y" — y' 



3. To find the distance between two points x'y' and x"y'' on the 

 sphere. The formulas for this, being well known, may be simply writ- 

 ten out for reference. If q denotes the distance, they are 



1 _|_ x'x" -f y'y" 



cos Q 



s'm Q = ± 



_J_ [(1 + ^/2 4.^2,(1 4. ^//2 + y/2)jJ' 



-(x" — a:')2 4- (;/" — y'f + {x'y" — x")/')^- 



tan Q = 



j[_ l(x" — x'y- 4- jy" — y')^ + jx'f," — x"y')2]i 

 1 -I- x'x'' -f y'y" 



4. A spherical conic is the intersection of a unit-sphere with a cone 

 of the second degree, whose vertex is at the centre of the sphere. The 

 arcs in which the cyclic planes of the cone cut the sphere are called 



