PHILOSOPHICAL SOCIETY OF WASHINGTON. 5t 



tan|-A. We might obviously have selected any other trigono- 

 metrical function, but this seems to lead to as simple a result as 

 4iiiy. 



If we make sin-|-A our unknown quantity our equation will be 



|4{(l-|^2)'+l}sin-A-4|(l-^^3)(l-^2^.) + 2}j 



I sin*iA — 4]l + ^v^2-^'[sin'^iA— 1^0, j 



and if we make sec ^ A the unknown quantity our equation will 

 be 



f sec* i A _ 2 (3 - Ax/ 8 ) sec' i A + 3 ( 6 _ '^-v' s+ \ 



Avhence it appearsthat the simplest equation is the one first ob- 

 tained in which the tangent is made the unknown quantity. 



Example. — Suppose a = 40 and /3 = 50. Then our equation 

 l)ecomes 



tan H A + tan ' ^ A + (- y/ 8 — 1 ) tan i A — i =. ; 

 ■whence by Horner's method 



tan i- A = 0.40788 15817 54736. 

 Whence A = 37° 03' 51".33 



B = 52° 56' 08".67, 

 a.nd the sides of the triangle are 



a = 35.807377 

 6 = 47.407275 

 c = 59.41058. 



SECOND SOLUTION. 



Let a, h, and c be the sides of the triangle opposite A, B, and 

 C respectively, and o and /3 as before; then we have (Fig. 1) 



_=cos ^ A ; whence— ;^-= 2 cos'-' -^ A = 1 + cos A =: 1 -j- ; 



a a C 



