266 ROYAL SOCIETY OF CANADA 



It also could be used to establish the theorem on polars, but though 

 perfectly symmetrical the formula is somewhat cumbersome. 



XXIII. Time will not be taken up with the application of this 

 formula to determine the tangents, poles and polars of the spherical 

 parabola, spherical ellipse or spherical hyperbola, though it will be 

 readily seen that in the case of the spherical parabola, at least, these 

 would be easily found. 



XXIV. The differential equation of the loxodrome gives the 

 equation of its tangent at once since 



d tan <p _ tan 6 ion (p + cot a \^ 1 + tan' 6 + tan' <f) ^ 

 dtan 6 " 1 + tan' 6 



tan y — tan <f> 



tan X — tan d 



XXV. At this stage we shall insert the proof for the condition 

 of perpendicularity of two great circles, 



^ tan ^ + 7i tan </) + C = o. 

 A'ian 6 + B' tan (f) + C" = o. 



Where a is an angle employed in the loxodrome, and making the 

 convenient substitution, to save writing, we have 



4 {I + &') + d4> 



— cot a = 



- cot (-^ + a) = A 



That is 



(1 + ^' + 0') 4- (^(1 + d') + «/>) (^(1 + &^) + dcf>) = o 



The elimination of 6 and <f) from this equation and the initial two 

 above gives the required condition for perpendicularity. 



A ^ t A ^ ,\ A CO A - C e 



^ 6 {~-r ^- 



^j = B - -IT 



B ' \ li ^ J B n li 



., A d + C . A' d + C 

 ^ = B- W 



This gives then 



BB' (1 -\- d') + (^ ^ + C) {A' 6 + 6") + {A - C 0) {A' - C 6) = o, 



