PARTICULARLY APPLICABLE TO SOME GEODETICAL PROBLEMS. 115 



sill a COS 7 +COS a sin 7 = —sill /3, 

 sin a cos /3 + cos a sin /3 = — sin 7, 

 sin a cos A — cos a sin ^ = sin (a — A), 



we obtain 



af sin a + xy sin ^ + a-s; sin 7 = ic sin (« — ^). 



Hence we derive this elegant theorem, 



• r. ■ be . , 



X sin a + y sin /5 + s sin 7 = — sin (a — A). 



11. From the form of the function which is the first member of 

 this equation, it will remain the same, although we change the angles 

 A and a into B and (i\ or into C and 7, provided corresponding 

 changes are made in the lines a, b, c and x,y,%: so that, on the whole, 

 we may conclude that 



X sin a + y sin /3 + s sin 7 



be . . 

 = — sm (a 



X 



A) 



= — sin (/3 - B) 



y 



ab . , _ 



= — sm (7 - C). 



This is the property of the figure which I proposed to investigate; 

 and it manifestly comprehends this other property, 



ax _ by cz 



sin (a - A) ~ sin {(i — B) ~ sin (7 - C) ' 



from which it also follows that 



X b sm {a — A) 



y ~ a ' sin (/3 - ^) ' 



X _c^ sin (a - A) _ 



~ " « ■ sin (7 - C) ' 



y _ c sin (/3 - B) 

 z "" A ■ sin (7 — C)' 



