traced upon the Surface of the Sphere. 411 



P. 284, Eq. (7.) may be modified in the same manner as (iv. 10.) was 

 modified. 



285, Eq. (10). This may be written 



cot d> = ~ cot a ' sin P'~ ~ cot A ' sin K '~ 

 sin /?, K 



and thus agree with a circle passing through a, ft, and K, A, ; as it should do 

 (see iv. 10.) the perpendicular to any circle passing through its spherical 

 centre. 



288, Prop. IX. In equation (4.) put 



[and from equation (2.), cos/? K = cot a cot A..., (4.)] 



Ib. In passing from the first to the second value of cos K, we have 



cos K = cos/3 cot a cot X + sin (3\/ 1 cot 2 a cot 2 A. 



_ cos /3 cot a sin a sin e + sin (3 \/ 1 sin 2 |ct sin 2 6 cot 2 a sin 2 a si 

 V ! sin2 a si " 2 e 1 sin2 sin 2 c 



sin a f 

 V ! sin2 a si " 2 e V 



cos jS cos a sin e + sin /3 x (+- cos e) 

 sin 2 e 



Now, in this, the sign of the denominator is , but independent of the 

 sign of the radical in the numerator of the fraction, and hence the double 

 sign db in the numerator ought to be retained in any result that flows from 

 our inquiries, though the denominator itself should be eliminated from the 

 expression. The same rule is true for the expression sin K, which gives 



_ sin /3 cos a sin 6 cos p* x ( cos e) 



sin K , / . . (8. j 



ziz V 1 sln a sin* f 



The equation of the circle then becomes 



( cos /3 cos a sin e + sin /3 (-+- cos e) } 



COS 



_i_ v i sin 2 sin 2 a ( \J 1 sin 2 e sin 2 a 



COt (D *^~ : : X / _ 



I . /, ( sin p cos a sin 6 cos p (~H cos 

 1+ sin f 



rty' 1 sin 2 sin* a 



from which the \/ 1 sin* e sin 2 a, which is common to both numerator 

 and denominator, vanishes of itself, its sign included. The equation then 

 reduces itself to 



VOL. XII. PART II. 3 G 



