778 Area, Spherical and Plane. [No. 128. 



by actual multiplication and reduction of terms with common factors 



this becomes 



. E ab . n f . , a 2 + b 2 ab n . 60* + 5a« 6« + 66* 



tan — = ~ sin C < 1 -f — — cos C + X \ 



2 4 I 12 4 720 



24 16 7 



E . E E 3 



For tan — substitute its value in arc — f- — + &c. and transpose all 



the terms after the first, then substituting for them their values in powers 



of the quantity on the right hand side, we shall have 



E 3 1 /ab\ 3 . o n , o ab . n /a*b 2 a*b% ... , \ 



— = — /— sm 3 G + &c. = — sm C ■ cos 2 C + &c. 



24 3|4/ 4 \ 48 48 / 



incorporating these terms, and multiplying the whole by 2, we have 



ab f a« + & ab nn r 



E^^sinC li + ^^-^cosC 



3a*-5^ft» + 3^ _ «lM^ii cosC + ^1 cos^ C + &c. ] 

 + 360 24 12 J 



The first term is the same as that for the area of a plane triangle 

 having the same sides and contained angle : the following terms there- 

 fore shew the difference between the areas of the two triangles. Of 

 these, we may take account of as many as suits our object; but in 

 ordinary cases it will be needless to regard any beyond the two first. 

 Limiting ourselves to these, the difference between the areas of the plane 

 and spherical triangles corresponds to an excess represented by 



a i sinC /^l±l 2 _icosC)orby l 6 sin C (a 2 + |»- 3 ab cos c) 



This expression shews that when cos C becomes — , or when C ex- 

 ceeds a right angle, the spherical area must exceed that of the plane 

 triangle. When the two terms within the brackets cancel each other, 

 the two triangles will have equal areas; and when the second term 

 exceeds the first, the spherical area will be less than that of the plane 

 triangle. 



The limits are easily assigned. 



The sum of a and b being given, a 2 -f- b® is a minimum, and 3 a b is a 

 maximum when a = b. In this case the triangles are isosceles, and 

 a 2 -f W- = 2 a 2 , and 3 a b = 3 a 2 ; hence the terms within the brackets 

 will cancel each other when cos C = | or when C = 48° 11' 23". 

 For equal areas this is the maximum of C. With isosceles triangles, if 



