PARTICULARLY APPLICABLE TO SOME GEODETICAL PROBLEMS. 119 



^ i" ■ ^' ■ n C . 1 a' . ^,\ 



or {- sm a + Y sin /i + - sm 7} = —be sin A 

 \a b c '] a 



/ |- sin a + ^ sin /3 + - sin 7! = j-acsm Rj V. 



^ f^ • ^ ■ n C . 1 c , . „, 



x^ <— sin a + Y- sm ii ^ — sin 7> = — «6 sin C 



By changing x, y, z, a, b, c. A' into x , ij , z, a, h, c, A and the contrary, 

 these formulfe serve for the conjugate triangle A'B'C . 



17. Another expression analogous to that found may be had by 

 substituting for y and s their values in the formula 



yz sin a + xz sin /3 + jry sin 7 = be sin A. 



By proceeding as before, we obtain 



, (« . b . c . '\ a b c T . -1 



X- { - sm a + j7 sm /3 + - sm 7} = - . j7 . - 6c sin ^ 

 la b c '] a b c 



Aa . b . ^ c . \ b' a c . „ ,^^ 



y- \- sin a + psm /3 + - sin 7) = -. -. -;ac sin ^> VI. 



\a b "^ c \ b a c [ 



Aa . b . ^ c . 1 



z- {— sm a + 7-, sin ii + -sm 7} 



\a be'] 



cab 

 c ' a''b' 



ab sin C 



It is remarkable, that the coefficients of sin a, sin /3, sin 7 in these 

 formulte are the reciprocals of their coefficients in the preceding. 



18. Other values of x" may be obtained by putting the values of 

 y and s in terms of x in the formulfe 



X- — 2xy cos 7 + y* = c^ 



x' - 2xz cos /3 + »" = b', 



y — Sys cos a + r = «■'. 



Of these I shall only put down that deduced from the last, as the 

 most symmetrical, 



