780 



Area, Spherical and Plane. 



[No. 128. 



excess of about 39". I have had one observed by day-light on which the 



excess was about 40". 5. The least side was 80 and the largest 92 miles. 



2 



Such a triangle does not often occur, but even this had about — only of 



o 

 the area of that on which the difference has been shewn to be utterly 



insensible. 



But as the greatest difference occurs when C exceeds a right angle, 



we may find the particular angle giving a maximum difference of area 



by making — — / (a? -J- ft* ) sin C — Sab sin C cos C I a maximum. 

 By differentiating, we have 



2- - / (a 2 + ft* ) cos C — 3 a b cos 2 C "I d C = o 



a 2 + ft* cos 2 C 



whence the maximum corresponds to — - — = — = — — 



3 ab cos C 



This hardly admits of being solved directly, but the indirect solution 



is very easy. 



Since C must be greater than a right angle, we may put C = 90 + % 9 



cos 2 C cos 2 Y _ _ a* 4- ft* . , 



is always + , it is plain 



whence 



cos 2 x m a v + 



-: and since 



cos C sin X 3 aft 



that X cannot be less than O nor exceed 45°. Hence the quantity 



cos 2 *v 

 . y will pass through all its values from o to 00 in every half 



quadrant. 



By tabulating this, as 

 in the margin, for every 

 degree of \, we may rea- 

 dily find, for any given 

 ratio of the sides, the ap- 

 proximate angle giving a 

 maximum difference of 

 areas. 



By means of this and 

 the former Table, it will 

 appear, that with equal 

 sides the angle of maxi- 

 mum difference of areas is 

 somewhat greater than 124°, and by a nearer computation the exact 

 value of C will be found 124°* 02''35", being the greatest angle, giving a 



c 



r-j «'+ 62 



c 



w* + * 



C 



& Sab 





3 ab 





° Sab 





90 



+ CO 



105 



0-52453 



120 



0-00000 



91 



1-75788 



106 



•48808 



121 



995977 



92 



•45612 



107 



•45264 



122 



•91763 



93 



•27881 



108 



•41798 



123 



•87320 



94i 



15217 



109 



•38389 



124 



•82601 



95 



•05306 



110 



0-35020 



125 



9-77546 



96 



0-97117 



111 



•31674 



126 



•72076 



97 



•90101 



112 



•28336 



127 



•66088 



98 



•83929 



113 



•24989 



128 



•59433 



99 



'78387 



114 



'21620 



129 



•51901 



100 



•73332 



115 



0*18212 



130 



9-43160 



101 



0-68657 



116 



•14750 



131 



•32661 



102 



•6*285 



117 



•11217 



132 



•19372 



103 



•60157 



118 



•07595 



133 



8*00980 



104 



•56226 



119 



•03864 



134 



7-70105 



105 



•052453 



120 



0-00000 



135 



— 00 



