20 Prof. J. Thomson on the true and extended 



2. What has been thus established aflPords an explanation 

 of the double sign + , before the expressions for the sine and 

 cosine of half any angle of a spherical triangle. Thus, in the 

 formula 



si„.A=+ / ^i"(^-^)sin(.-c) 

 ^ ~~ t^J sm sin c 



if we denote the value of ^ A corresponding to the sign +, 

 by ^ A', we shall have the other, corresponding to — , equa 

 to — ^ A'; whence the values of A are A' and —A'; or, by 

 adding 2 tt to the latter, according to the preceding note, the 

 values will be A'and2 7r— A'. We see, therefore, that the 

 positive value of sin ^ A gives the value of | A in the less of 

 the two triangles bounded by a, /;, r; and the negative value 

 that of I A in the greater*. 



3. In like manner, in the formula 



^ —V sm o sm c 



the positive and negative values will give respectively ^ A 

 equal to | A' and tt — ^ A'; and consequently A equal X,o A! 

 and 2 tt — A', as before: and, in a similar manner, we may ex- 

 plain the double sign in the formulae 



'sin (5—-^) sin(5~c) 



tanJA=±Y/- 



sin s sin {s—a) 



, . . 2 \^ s\n s sm I s—a) sm is— b) ^m is— c) 



and sin A = H ^^^ — ^, — ^ ^^ -' 



— sin sin c 



4. By considering in a similar manner any of the formulae 

 which determine the sides bv means of the angles, we shall 

 arrive at other results which do not seem to have been hitherto 

 observed. Thus, from the formula 



cos A + cos B cos C 



cos a = ; :^r—. — ^ , 



sm B sin C 



and the corresponding ones for cos h and cos c, we see that 

 when the three angles are given, each of the sides has two 

 values of the forms a' and ^v—a, h' and 2 7r — 6', c and 

 2^— c'. The values a\b\c\ each less than a semicircle, 



* The same conclusions might be derived from the formula for sin ^ A, 

 taken, not with both its signs, but with either. For, since sin Q =: sin (^ — Q), 

 we should have for the value of \ A, corresponding to the positive sign, 

 either ^ A' or -jr — ^ A' ; and consequently A equal to A' or 2 Tr—PJy as 

 'ore; while the negative value will give — ^A' and — cr-f ^A'j by doubling 

 [ch, and adding to each 2 t, we get 2 tt— A' and A', as before. The 

 also be done by means of the formula for cos \ A. 



