33.

The quantities p, q, r, s are respectively equal to P, Q, R, S.

(A + B) and (a + b), are each less than π,

(A – B) and (a - b), are each less than
1⁄2

and are of the same sign,

C

2

π

and are each less than ;

π

p, r, s, P, R, S are all positive quantities.

COR. From (viii.) it appears that cos(A+B) and cos (a + b) are of the same sign, and therefore 1⁄2 (A + B) and †(a + b) are both greater or both less than T.

П

2

DEF. When two angles are both greater or both less than a right angle, they are said to be of the same, or of like, affection.

Thus (A + B) and (a + b) are of like affection,—a property of spherical triangles which may be added to those enumerated in Art. 27.

These four equations are called Napier's analogies, are of great use in the solution of triangles*.

and

*The quantities p, q, r, s, P, Q, R, S, will be most easily remembered from Napier's first and second analogies written thus,

35.

The following expressions can, without much difficulty, be proved from the formulæ of Art. 33; we will however establish them from the fundamental formulæ (i), (ii), (iii).

And if S=(a+b+c), S-a= 1 (a + b + c ) − a = {(b + c − a),

2

sin b. sinc

{sin S. sin (S-a). sin (S−b). sin (S−c)}..(xviii.)

The positive signs of the square roots are taken in these

2

sin B.sinC/{-cos,S". cos(S– 4).cos(S′ – B). cos(S′-C)}(xxiii)

*D