208 Equations of Condition for a Quadrilateral. [No. 123. 



E ab . ^ ( a* + b* 



c ^ • • i .. , ^ a0 . „ ( a* -\- b* 

 of the original equation, becomes, tan ~ — -j- sin C < 1 + -jx + 



1 f ab „ I « 2 + b 2 



. j j 1 -^ cosC (V+ -J2- + &C.) 



6 «4 -J. 5 a 2 b 2 _f 6 64 1 r « 6 / «2 _l_ b 2 

 + &C ' 



720 



aP-b* 



a 1 b* / a 2 + 62 a 3 7,3 . % v 



+ 16 -cos2 C(l + — q + &c.) —_ _cos3 C(l + &c.) + &c. J by 



actual multiplication and reduction of terms with common factors, this 

 becomes 



fo„ E a h • n ( . , a2 + b2 ab n 6 « 4 + 5 a 2 52 + £4 



tan <-> = j- sin C { 1 H — cos C + - — 



2 4 I 12 4 720 



a 3 6 + a 6 3 a 2 b 2 » 



-cos C + -Jq— cos2 C + &c. J 



E F F3 



3. For tan ~ substitute its value in arc ^ -4- -57- + &c. and transpose 



E 3 



24 "*" ^ C *' anc * su kstituting f° r them their values in powers of the right 



E 3 1 /» 6 v 3 a 6 



hand quantity, — = >- ^— J sin 3 C + &c. = -r sin C 



/«2 yi «2 62 



V ^g— — -|g- cos 2 C -f- &c. J then incorporating this and multiplying 



the whole by 2, we have 



ab . f «2 + &2 «& 3«4 — 5 «2^2 +3 64 



E = ~^~ sinC -! 1 + — f^ — — -7- cos C -\ 



2 J n 12 4 T 36O 



«3 6 + a 6 3 a 2 62 



cos C + ... - cos 2 C+&C 



24 n 12 



■} 



4. 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 therefore 

 shew the difference between the areas of the two triangles. Of these terms 

 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 



_ , a 6 /a 2 + b 2 ab \ 



corresponds to an excess represented by -j- sin C \ — ,» — —r~ cos C) 



or by -^j- sii\ C I a 2 + & — 3 a b cos C. 



, ab . r 



by -^- sin C J 



5. This expression shews that when C exceeds a right angle (cos C be- 

 coming — ) the spherical area must exceed that of the plane triangle. 

 When the two terms within the brackets cancel each other, the two trian- 

 gles have equal areas ; and when the second term exceeds the first, the 

 spherical area will be less than that of the plane triangle. 



6. The limits are easily assigned. 



7. The sum of a and 6 being given, a 2 + 6^ is a minimum, and a 6 or 



3 a 6 is a maximum when a — b. In this case the triangles are isosceles, 

 and a 2 + 6 2 =■ 2 a 2 , and 3 a 6 = 3 a 2 ; hence the terms within the brack- 

 ets will cancel each other when cos C = J, or when C =? 48° 11' 23". 

 This for equal areas is the maximum of C. With isosceles triangles, if C be 

 less than this, the spherical area will be less than that of the plane triangle. 



