280 Mr DAVIES on the Equations of Loci 



VI. 



Required the co-ordinates of the points of intersection of two given great 



circles. 



Denote the given circles by 



cot< = tan \ cos (6 /c,) .......................... .................. (1.) 



cot(f) = tan X,, cos (0 ) ................. , ..... ... ................. (2.) 



From these we have 



tan \ _ cos (6 K ;/ ) _ cos K,, + sin K,, tan 6 

 tanX /7 ~~ cos (6 /c,) ~ cos K, + sin K, tan 6 ' 



cos K / tan X // 



^ 

 tan X x sin K, tan X,, sin K,'" 



. n tan X. cos K. tan X,. cos K.. ,. ,. 



im r H 7 "^ ........... (4.) 



V tan i X / 2 tan X/ tan X y/ cos (K,, K t ) + tan 2 X /7 



co:)Q=: tan \sinK, tan \ sin /c /; . g 



\/ tan *X y 2 tan X, tan X,, cos (* /c y ) + tan \' ' ' 



Again, from equations (1, 2) we find 



6 = cos- l ( cot (p cot X,) + K, 

 6 = cos-^ cot cot X /; ) -f- /c /; 



and therefore, 



cos - 1 ( cot (j) cot \) cos- '(cot (J) cot X /7 ) = K,, K f 



Or taking cosine-function of both sides, transposing and squaring, we get 



I sin />>"""*""/*. . j. 



" ' (6.) 



k _ 



V cot *X, 2 cot X, cot X,, cos K U K, + cot Z X A 



