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SPHEEICAL TEIGONOMETEY AND 

 PLANE ASTRONOMY. 



TT 7T 



1613. In a spherical triangle ABC, a = b = -z, c = 9 j prove 



v 



that the spherical excess is cos" 1 - . 



y 



1614. In an equilateral spherical triangle ABC, A', B', C' are 

 the middle points of the sides : prove that 



BC BC 



2 sm = tan -^- . 



'-i A 



1615. In an equilateral spherical triangle, whose sides are 



each a and angles A, 2 cos s sin 5- = 1. 



l - 



1616. ABC is a spherical triangle, each of whose sides is a 

 quadrant, P any point within the triangle : prove that 



cos 2 AP + cos 2 BP + cos 2 CP = 1 ; 

 cos A P cos BP cos CP + cot BPC cot CPA cot APB = ; 



and that tan BCP tan CAP tan ABP = 1. 



1617. A point P is taken within a spherical triangle ABC 

 whose sides are all quadrants, and another triangle is described 

 whose sides are equal to 2 A P, 2BP, 2CP respectively: prove that 

 the area of the latter triangle is twice that of the former. 



1618. A spherical triangle ABC is equal and similar to its 

 polar triangle : prove that 



sec* A + sec 2 /? + sec 2 C + 2 sec A sec B sec C = 1. 



1619. If the sum of the sides of a spherical triangle be given, 

 its area is greatest when the triangle is equilateral. 



