1842.] 



Equations of Condition for a Quadrilateral. 



209 



8. Again when cos C is a maximum, C 



: In this case, a? -J- W> = 3 a b 

 b 3 + \/!T_ 



or 1 4- >t = 3 ~* ; the solution of a quadratic will give 



a 2 a a z 



2-618 nearly. This is the maximum inequality in the sides so as to have 

 equal areas. 



9. In like manner may be found the value of the angle for any given 

 ratio of the sides within these limits ; or the angle being given, the ratio 

 of the sides may be found. 



10. The following Table shews for given ratios of a and b the value of C 

 giving equal areas : — 



11. If the sides were so large in regard 

 to the radius, as that the terms omitted 

 could sensibly affect these results, it would 

 be necessary to take those of the next, and 

 perhaps also of higher orders. 



12. To ascertain the actual difference in 

 the areas of the spherical and plane tri- 

 angles in an extreme case, suppose an 

 equilateral with sides of 1 § degrees : the 

 direct formula gives the excess = 61.217; 

 and the difference in the areas of the two 

 triangles will be -3951 square miles, cor- 

 responding to an excess of 0-"005245. One- 

 third of this would be the difference on 

 each angle, and were it ten times as great, 

 it would still be, in Troughton's phrase, a 

 quantity less than what is visible in the 

 telescope. 



13. It is almost needless to remark, that 



the supposed triangle is larger than any 



which has yet occurred in practice. The great 



triangle in the French arc, long supposed 



to be the largest in the world, has an excess 



of about 39". I have had one observed 



by day-light on which the excess was 



about 41".5. This least side was 80 miles, and the largest 92-6. Such a 

 triangle does not often occur, but even this has only about § of the area 

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



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

 we may find the particular angle giving the maximum difference by 



b 



a 



CosC 



Log. cos C 



C 



10 

 11 



12 

 13 



200 



300 

 921 



330 

 244 



360 

 269 



9-82391 

 •82588 

 •83109 

 •83869 



48.°11 / 

 47-57 

 47-20 

 46-23 



14 



296 

 420 



•84804 



45-11 



1-5 



325 

 450 



•85867 



43*46 



16 



356 

 480 



•87021 



42-08 



17 



389 

 510 



.88238 



40-18 



1-8 



424 

 540 



•89498 



38-16 



1-9 



461 

 570 



•90783 



3601 



20 



500 

 600 



•92082 



3333 



21 



541 

 630 



•93386 



30-50 



22 



584 

 "660 



•94687 



27-46 



2-3 



629 

 690 



•95980 



24-16 



2-4 



676 

 720 



•97262 



20*08 



25 



725 

 750 



•98528 



14-50 



26 



776 

 780 



•99777 



5-48 



making—; \ (cfi + W \ sin C — 3 a b sin C cos c| 



differentiating, we have 



a b 

 24 



{ (« 2 + b ) 



cos C — 3 a b cos 2 C 



a maximum : by 

 \dC=zo: 



