traced upon the Surface of the Sphere. 283 



which agrees with the expression obtained in (VII, 6), bearing in mind that 

 8 is put for e / and K, for & p and . for /?, + =. 



Cor. 2. If both be meridians, then cos d =. cos K.,, K /} that is, the arc of 

 the equator cut off by them measures their inclination, as we know it should. 



IX. 



To find the equation of a great circle having a common intersection 

 with, two given circles. 



Let cos p t = cos A, cos (j) + sin A, sin tp cos 6 K, (1.) 



cos p a = cos \ cos < + sin A,, sin < cos Q K U 



be the given circles ; and let the great circle sought be denoted by 



cot</> = tanAcos$ K (3.) 



in which we have to determine A and K. 



Multiply (1) by cos A, and (2) by cos A /7 and subtract : then 



S i n _ sin \ cos \ cos 6 K, cos X, sin \, cos 6 K a 

 cos p, cos \ cos p u cos A, 



Multiply (1) by sin \ cos Q K,, and (2) by sin \ cos ^ ^, and sub- 

 tract ; which gives 



, _ cos p, sin \ cos 6 K a cos p a sin A ; cos 6 K, ..,,, 



cos A, sin A /y cos 6 K" cos \ sin A, cos 6 K, 



Divide (5) by (4) ; then we find the equation of the sought circle : 



cos p, sin A ;; cos 6 K a cos p a sin \ cos 6 /c ; ,~^ 

 cot T* cos p, cos A /; cos p H cos A, 



Or, to accommodate it to the form (3), we must expand the cosines of this, 

 and we find 



