210 



Equations of Condition for a Quadrilateral. 



[No. 123. 



the maximum corresponds to 



«2 + &2 C os 2 C 



This is an equation 

 3 a b cos C ^ 



which scarcely admits of a direct solution, but the indirect solution is 

 very easy. 



15. As C is to be greater than a right angle, we may put 90 -f- X = C 



. cos 2 c _ cos 2X 

 '' cosC "" s i n x 



As — — - — is always +, it is plain that X can- 

 o a b 



not exceed 45°, nor be less than 0. Hence the quantity ^21 — ^ will pass 



sin X 

 through all its values from to 00 every half quadrant. By tabulating 



this, as under, for every degree of X> we shall have by inspection for any 

 ratio of the sides, the approximate angle giving a maximum difference 

 of areas. A nearer approximation may be got by making proportion for 

 the differences between the tabular and actual quantities in the usual way ; 

 and by computing another value on each side of the angle so found, we may 

 by successive steps bring the approximation as close as we please. 



16. By means of this and the former 

 Table, it appears that with equal 

 sides the angle of maximum differ- 

 ence of areas is somewhat greater 

 than 124°, and by another computa- 

 tion it will be found that the exact 

 value is 124°-02'-35" being the great- 

 est angle giving a maximum differ- 

 ence of areas. For any other ratio 

 of sides the angle will be smaller. 



For the ratio 3 -J- \/ 5 the angle is 

 ~~ 2 





u,o 





OiU 





u 



o 





" u 





■MS 





<M 



in 





a> o 





M 8 





to 



o 



c 



§|° 



C 





C 



o 

 o 



w 





bb 





bb 





bb 





o 





3 





o 



91 



1-75788 



106 



0-48808 



121 



9-95977 



92 



•45612 



107 



•45264 



122 



•91763 



93 



•27881 



108 



•41798 



123 



•87320 



94 



•15217 



109 



•38389 



124 



•82601 



95 



•05306 



110 



•35020 



125 



•77546 



96 



0-97117 



111 



0-31674 



J 26 



9-72076 



97 



■90101 



112 



•28336 



127 



•66088 



98 



•83929 



113 



•24989 



128 



•59433 



99 



•78387 



114 



•21620 



129 



•51901 



100 



•73332 



115 



•18212 



130 



•43160 



101 



0-68657 



116 



0*14750 



131 



9-32661 



102 



•64285 



117 



•11217 



132 



•19372 



103 



•60157 



118 



•07595 



133 



8-00980 



104 



•56226 



119 



•03864 



134 



7-70105 



105 



•52453 



120 



•ooooo 



135 



-00 

















10 



1 



«2 + 



62 g 101 Cos2C 



3 a b~~ 1S ~30~ andLo S- Cos c is 0-52720, which corresponds to an angle 



of about 4'-25 less than 105, or 104 o -55« , 75 ; and so in other cases. When 

 the ratio of the sides becomes indefinitely great, the maximum difference 

 angle approaches indefinitely near 90. 



17. In well chosen triangles, there are not usually any very great differ- 

 ences in the sides, and hence practically the greatest differences will usually 

 occur when C is not far from 120°. 



18. If for example we suppose a triangle with sides of a degree each, and 

 containing an angle of 120°, by the original formula the excess is 27"-210 

 and the difference in area between the spherical and plane triangles is 

 0-18214 square miles, corresponding to an excess of 0"-0024176. On a 

 triangle with degree sides and the maximum angle of 124°02 / -35" the 

 excess is 26"-035 the differences of areas 0-18320 square miles, corres- 

 ponding to an excess of 0"-0024318. Such differences though utterly in- 



