any whole number or zero, and admitting of different values in each of the parts. Let us, therefore, substitute in (7) the following values of these parts:— We shall have for the factors of the first members values similar to the following: m m cos [(m ̧ + m ̧) x + } (A + B)] = (− 1) 1 + 2 cos ↓ (A + B) ገቢ sin [(m, + m„) = + ↓ (A + B)] = (− 1)”1 +TMa sin ↓ (A + B) } cos [(m, · m1) π + § (A — B)] = (— 1) 1 2 cos(AB) m sin [(m, —m ̧) x + ↓ (A — B)] = (— 1) "1 — "? sin } (A — B) Now whatever the values of m, and m2 m1 + me and m1· , m2 are both even or both odd at the same time, and therefore the above substitution gives the same sign to all the first members of our equations. In the same way it is shown that the second members will all have the same sign; and we may consequently express the result of the substitution thus : cose sin (A + B) = (— 1)" cos c cos (a — b) which single group involves both (7) and (8). The group (7) will represent one series of triangles, while the group (8) will represent another series, the two differing in each of their elements by some multiple of 2x, and the primitive triangle may belong to one or the other of these series. We may dispense, therefore, in practice, with group (8), by deducting 360° or 2′′ from the elements found by group (7), till they become less than 360°, as required for use. (See top of this page.) 6. When the parts of the triangle are interchanged in GAUSS's equations, it would seem to require proof that the same sign, whether + or -, must continue in these equations; i. e. that when the triangle is such as to satisfy the equation, cosc sin (A+B) = COSC cos (a - b) * Apply (3) and (6) of Art. 70 to sin results. Or observe that cos (n« + † c) · and cos of 2 n3 +c to obtain these cosc according as n is even or odd. == To demonstrate this, we will show that the groups to which the equations (e) belong may be derived from (7), and those to which (ƒ) belong from (8), by merely linear transformations, and therefore without again introducing the double sign. Let the equations (7) be written thus:— The sum and difference of the first two, and the sum and difference of the last two, give* COS (A+B+c) = COS (ABC) sin c sina sin b cos c cosa cos b cos c By differently combining these four equations, two and two, we may either reproduce the group (g) or the two groups represented by (e). Thus the sum and difference of the first and third, and of the second and fourth, give Precisely the same transformations applied to (8) would of course give a similar result with the negative sign. Hence In the three groups which GAUSS's equations form by the permutation of the letters, the positive sign must be taken in all the equations, or the negative sign in all of them. AUXILIARY ANGLES. It will be convenient to premise here the following proposition, upon which depends the proper employment of auxiliary angles in preparing our general formulas for logarithmic computation. In the equations k sin m = (9) whatever the values of m and n, we can always determine k and so as to satisfy at once these equations, and any one of the following conditions arbitrarily imposed. The six conditions above stated are obviously equivalent to the following; 1st, k+; 2d, k―; 3d, sin +; 4th, sin ; 5th, cos +; 6th, cos -. Of these six conditions, however, we commonly employ only the first, third, or fifth. m The quotient of the equations (9), tan = gives two values of under 360°.* n Hence, also, two values of k, which will be numerically equal with opposite signs, * Because for any arc less than 360° there is always another arc < 360°, having the same tangent. If the former be in the first quadrant, the latter is in the 3d; if the former be in the 2d, the latter is in the 4th quadrant. since the two values of sin & will be numerically equal with opposite signs, as also the two values of cos p. If we restrict the sign of any one of the three quantities k, sin p, cos p, the signs of the other two will become known, and there will be but one value of k and one of under that restriction. SOLUTION OF THE SEVERAL CASES OF THE GENERAL SPHERICAL TRIANGLE. In the solutions of the various cases of spherical triangles, it is of the first importance to have simple and clear precepts, both for removing the ambiguity that occurs in every case and for determining properly the auxiliary angles. Examples might be pointed out, in recent works on trigonometry, of incorrect numerical solutions resulting from an erroneous application of precepts, in themselves correct, but not sufficiently simple or explicit. I have, therefore, given special attention to this point in arranging the following solutions. These solutions have also been carefully verified by the computation of the two triangles following: The first of these triangles requires the positive sign in GAUSS's Equations, and the second requires the negative sign. 1. Given b, c, and s, to find a, в, and c. The general relations between the given and required parts are cos a = cos c cos b+ sin c sin b cos A sin a cos B = sin c cos b. -cos e sin b cos A sin a sin B = sin B sin A (1) and similar forms to the last two, with c and c interchanged with B and b. The second members being computed, the numerical value and the sign of cos a will be determined from the first equation. From the second and third sin a and B are determined precisely as k and in the preceding section and are subject to the same ambiguity. The ambiguity will be removed, therefore, when the sign of either sin a, sin B, or cos в is given, and in like manner when the sign of either sin c or cos c (the other required parts) is given. The solution may be adapted for logarithmic computation, and the condition required for removing the ambiguity may be varied. Let k and be determined by the conditions (9), taking m = sin b cos a, and n = cos b, and adopting the first arbitrary condition; then these conditions together with equations (1) assume the following form: *For the second members of the 2d and 3d of (1) being computed, they will be fixed quantities like m and n in (9), and sin a occupies the place of k, and в that of in the same equations (9). Or, eliminating k and adopting the third condition imposed on eqs. (9) If the quadrant in which a is to be taken is given, then In (3) and (4) we may also limit to values numerically less than 90°, the sign of the tangent being determined according to the fifth arbitrary condition following (9). If both a and b are less than 180°, as not unfrequently happens in the applications of this problem, let then m and n are both positive (k being positive) and the following form may be employed :-§ Check. For the purpose of verification we may employ with (4) or (5), the formula sin a sin B = sin b sin a; and with any of the preceding solutions the following check : sin (c sin a cos B tan A * The value of k which appears in the 2d and 3d of (3) is obtained from the 2d |