A=103° 59′ 57′′, B = 46° 18′ 7′′, C= 36° 7′ 52′′; required the side a. a= 42° 8'48". 3. The three angles of a spherical triangle are 120° 43′ 37′′, 109° 55' 42", and 116° 38′ 33′′; required the three sides. 115° 13′ 26′′, 98° 21′ 40′′, and 109° 50′ 22′′. CASE III. (58.) Given two sides a, b, and the included angle C, to find the other parts. By Napier's analogies, cos. (a+b): cos. } (a ~ b) :: cot. C : tan. † (A + B) sin. (a+b): sin. § (a ~ b) : : cot. § C : tan. § (A ~ B). These serve to determine the angles A, B, opposite to the given sides; after which the third side c may be determined by either of the remaining two analogies of Napier, viz. cos. (A+B): cos. (A ~ B) :: tan. c: tan. § (a + b) K EXAMPLES. 1. In a spherical triangle are given a 38° 30′, b= 70°, and C = 31° 34′ 26′′, to find the other parts. 1. To find A and B. cos. § (a+b) 54° 15′ ar. comp. 0·2334015 ar. comp. sin. 0.0906719 (A+B) must be acute, because 1⁄2 (a + b) is acute. By taking the sum and difference of these results we have, B= 130° 3' 11", and A = 30° 28′ 11′′. * There will be no necessity to refer to the tables for the tangent of this arc, we shall obtain it by subtracting the right-hand arithmetical complement in the preceding logarithmic process from that on the left, adding 10 to the index. For calling the right-hand complement p, and the left q, and recollecting that log. tan. = 10 + log. sin. —log. cos. = 10+ (10 −p) — (10 — q), we have log. tan. 10+q-p. When, in the case we are considering, the only part required happens to be the side opposite the given angle, the finding of the other two angles then becomes merely a subsidiary operation, and the determination of the required side, by Napier's analogies, seems somewhat lengthy. But a shorter method of solution is deducible from the fundamental formula, ... • (1). cos. c = cos. a cos. b + sin. a sin. b cos. C For substituting cos. a tan. a for its equal sin. a it becomes cos. ccos.a (cos. b + tan. a sin. b cos. C). Hence, to find the side c, we must first determine a subsidiary angle from the equation 2. The same parts being, given as in the last example, to determine Other formulas for the determination of c might be easily deduced from the same equation (1), but this is as short and as convenient as any. We might also introduce here a distinct formula for the determination of one of the angles A, by help of a subsidiary arc w; but as little or nothing would be gained, in point of brevity, over the process by Napier's analogies, we shall not stop to investigate it. 3. In a spherical triangle are given a=43° 37′ 38′′, b=37° 10′, C=120° 59′ 46′′, to find the side c c68° 46′ 2′′. 4. In a spherical triangle are given the two sides, equal to 37° 10 and 68° 46′ 2′′, and the included angle equal to 39° 23′; required the other two angles. 33° 45′ 3′′ and 120° 59′ 49′′. 5. Given the two sides equal to 44° 13′ 45′′ and 84° 14′ 29′′, and the included angle equal to 36° 45′ 28′′; to find the other parts. The angles are 32° 26′ 6′′, and 130° 5′ 22′′, and the side 51° 6' 12". CASE IV. (59.) Given two angles A, B, and the interjacent side c, to find the other parts. The solution of this case, as well as the former, is comprehended in Napier's analogies; the one pair, viz. cos. (A+B): cos. † (A ~ B) :: tan. c: tan. 1⁄2 (a + b) c: tan. } (a —b); determining the unknown sides a, b, and either of the other pair, viz. cos. (a + b): cos. 1⁄2 (a ~ b) :: cot. 1⁄2 C : tan. § (A + B) sin. § (a + b) : sin. § (a ~b) :: cot. 1⁄2 C : tan. § (A — B) ; enabling us to find the unknown angle C. EXAMPLES. 1. In a spherical triangle are given two angles equal to 39° 23′, and 33° 45′ 3′′, and the interjacent side equal to 68° 46′ 2′′; to find the remaining parts. 1. To find the Sides. cos.(A+B)36°34′11′′ar.comp. 0·0951980 ar. comp. sin. 0.2249260 If the angle opposite to the given side be the only part required, a more compendious method of solution may be obtained by introducing Thus the formula (B) art. (50) be a subsidiary arc, as in last case. comes when cos. A tan. A is substituted for sin. A, cos. C=cos. A (tan. A sin. B cos. c — cos. B); The log. tangent of this arc will be equal to log. sin. before given, increased by 10. |