220 
ON SOME ANALYTICAL FORMULAE, AND THEIR APPLICATION 
[32 
It is hardly necessary to observe, that if 
+ B V + C£=0 
(75) , 
(76) , 
Ax + By + Cz = 0 
represent the same great circle, 
A : B : C = A + vB 4- gC : vA + B + : pA + \B + C 
(77) , 
(78) . 
A : B : C=aA + t)B + gC : f)A + t>B + {C : gA +fB +cC 
Inclination of two great Circles. 
Let the equations of these be 
f A£ + B v + C£ = 0 
[or Ax + By + Cz — 0 
f A'£ + B' v + C'£ = 0 
[ or A'x + B'y + G'z = 0 
(79) , 
(80) , 
(81), 
(82), 
and let e, e, have the same values as above, and e', e', corresponding ones. To obtain 
the inclination of the two circles, we have only, in the formulae given above for the 
distance of two points, to change p, p', o>, co', into e, e', e, e. 
The distance of a point from a given circle may be found with equal facility; for 
this is evidently the complement of the distance of the point from the pole of the 
circle. In like manner we may find the condition that two great circles intersect at 
right angles, &c. 
There are evidently a whole class of formulae, not by any means peculiar to the 
present system of coordinates, such as 
Ax + By + Gz — s (A'x + B'y + G'z) 
(83), 
for the equation of a great circle subjected to pass through the points of inter 
section of 
Ax + By + Cz = 0, A'x + B'y + G'z = 0. 
(84), 
x, y, z = 0 
a, b, c 
a', V, d 
Again, 
for the equation of the great circle which passes through the points given by the sine 
systems a : b : c and a' : b' : c', &c., and which are obtained so easily that it is not 
worth while writing down any more of them. 
Transformation of Coordinates. 
Let X u 7 lt Z ly be the new points of reference, and suppose X x is given by the 
conjoint systems e = a, b, c, e = a, /3, 7; and similarly Y 1} Z lt by the analogous systems