fBEATTY] AN OUTLINE OF ANALYTICAL SPHERICAL GEOMETRY 267 



or 



(1 4- 6^) {A A' + BE' + CC) = o. 



This requires then that 



A A' + BB' + CC = o, 



which is, therefore, the condition for perpendicularity of two great 

 circles. 



XXVI. The ecjuation of the great circle through the point 

 {d' (j)') and perpendicular to the great circle 



A tan d + B tan (j) + C = o, 



is at once seen to be 



tan /? tan ^ 1 



tan 0' tan </>' 1 



ABC 



An application of this would be to find the equation of the normal to 

 a curve at any point when the equation of the tangent is known. 



XXVII. It might seem at first glance that there is here no 

 analogue of the parallelism of plane geometry. However, we should 

 apparently be justified in taking any two great circles through a 

 point as parallel. It would be interesting to find the locus of the 

 points of bisection of parallel chords of a sphero-conic. Analogy 

 suggests that this locus would in every case be a great circle. 



XXVIII. The connection between 

 the two systems of co-ordinates is similar 

 to that in plane geometry. To express 

 X and 1/ in terms of 6 and r we have at 



once 

 P cos 6 tan r = tan x 



sin 6 tan r = tan y 

 That is, 



tan y 



tan d = 



tan X 



Therefore 



= tan' 



(tany\ 

 tan xj 



Besides 



tans r = tan'' x + tan* y. 



