1842.] Area, Spherical and Plane. 777 



sides are given, is greater or less according as the contained angle is 

 greater or less than 90. 



When C = 90 the equation becomes simply- 

 tan \ E = tan | a tan ^ b ; 

 and as every spherical triangle by letting fall a perpendicular becomes 

 the sum or difference of two right angled triangles, this expression may- 

 be extensively used. 



When the second term in the denominator becomes = 1, tan ^ E = 

 oo , whence \ E = 90° and E = 180°. 



When the second term exceeds unity, the whole expression becomes — , 

 hence \ E is in the second quadrant, and the excess exceeds 180°. 



In order to apply the expression above found to the comparison of the 

 area of the spherical with that of a plane triangle, it may be otherwise 

 written 



tan \ E = tan \ a tan 1 b sin C ( — = — -A 



\tan y a tan ^ b cos C/ 



when the denominator of the term within the parenthesis may be ex- 

 panded in the usual way. 



For tan £ a and tan l b substitute their values in arc to radius 1 by 

 the formula 



l 3 2 5 . 17 7 

 tanx = X + * X f5 X + 315 X + &c 



and we have tan % a tan i b = 



/i+ t + a * + l ? " 7 +&c w*+S + b " + 17 b7 \ 

 \2 24 240 39720 ' ) \2 3 240 39720 ) 



which by actual multiplication becomes 



4{ 1+ ~~12 + 720 



136 a* + 63 a* b 2 + 63a 2 6 4 + 136 6 6 &c l 

 40320 J 



This expression and its powers being substituted in the expansion of 

 the original equations gives 



. E ab f , , a 2 + b 2 , 6a 4 + 5a2 b 2 + 66 4 , « 1 



H** "T2 1 -- 720 +&C 'J 



j sin C 



cos 3 c(l + &c.)+&c. I 



o 3 6 3 



