320 Mr DAVIES on the Equations of Loci 



SUVK = 



sin a, sin a,, 

 which expanded, becomes 



(cota,, cot*-) (cot a, cotr) = cosec 1 <r sin 2 K (1.) 



Also, if for /3, and fi u in (9. IV.) we put K and + K, we shall readily 

 obtain the equation of the great circle MN; viz. 



cot0sin2/c = sin K + 6 cot a,, + sin K <9cota, (2.) 



Eliminating one of these cotangents, as cota a from (1.) (2.) we find 



cotd) sin/c coto- sin 6 (1 cosec 8 a- cos 2 K) sin K + 6 cot 



cot*a, 2cos/c - Z -- ___ - J - ' - - 



, 



sin K sin/c 



The differential of (3.), taken with respect to the arbitrary quantity cot a, , 

 gives the equation of the base consecutive to MN. It is 



cos K (cot d) sin K cot <r sin 6) 

 cota /= =- y . - - (4.) 



sin /f a 



Eliminating cot a, between (3.) and (4.) we obtain 



cos 2 K (cot sin K cot <r sin 0) 2 coto- cot sin 2/c sin K 6 = (1 cosec* * cos*/c) (cos 2 0-cos 2 K) ... (5.) 



Expand the vinculated quantities, and add (sin 2 /c cos 2 & sin l 0cos*/c)cot 2 ir 

 to both sides, and we shall find 



(cot (f)COSK COt a- COS 0) 2 = (COS 2 6 COS* K) COS6C 2 (6.) 



Or extracting and transposing, 



COS <r COS -+- VCOS 2 d COS 2 K ,* -^ 



cotcp = . (.') 



COS K Sin <r 



This is the equation of a circle, resolved according to cot <p *. Hence the 

 locus sought is a circle, as is well known from geometrical considerations. 



It may, however, be readily brought under the more familiar form (1. 1.) 

 as follows. Resume (6.), expand and multiply all the terms by sin* <r sin 2 <. 



* See Note (A) at the end. 



