and, performing an operation similar to the preceding, there will result, - A cos B 2 sin AB Sin (B + AB).... (2.) 6. To find the difference of the tangent of an arc or -angle, assume sin A sin B COS B and, operating as before, there will result, 8. For the difference of the secant, take first, (sec A sec B) COS B COS A: whence, form. u, chap. iv. sec A-sec B = 2 sin (AB) sin (A + B) COS A COS B Making in this equation substitutions analogous to the preceding, we shall have cing the requisite substitutions in the value of sin a → sin B, form. U, chap. iv. obtain A coseс B = 2 sin AB COS (B + LAB) sin B sin (BAB) .... (6.) 10. These six equations are rigorously correct whatever the magnitude of AB may be. Let us trace the modifications they will undergo when the variation becomes indefinitely minute, or AB becomes B. Returning to the first equation, we shall, by expanding cos (BAB) according to form. u, chap. iv. have Asin B2 sin AB (COS B COS AB sin B sin 4B) 2 sin AB COSB COS AB - 2 sin2 AB Sin B sin AB COS B sin B (1 COS AB). Now, if AB be indefinitely small, so as to approximate very nearly to evanescence, sin AB will also be indefinitely small, or practically evanescent, while cos AB will differ indefinitely little from radius. In that case the differential equation, will become sin B sin B cos B sin B (1 COS &B); which, since sin dB = JB, and cos B = 1, reduces to sin BB cos B.... (7.) 11. Proceeding similarly with equa. (2), we obtain cos BB sin B, or coS B—— JB sin B.... (8.) 12. In like manner from equa. 3 and 4, we obtain And so on, for the differentials of the secant and the cosecant. 13. Or, having found the differentials of the sine and cosine of B, others may Since versin B = 1 versin B be deduced thus: COS B, we have cos BB sin B .... (11.) 14. The differences and differentials of the principal lineo-angular quantities (chap. i. art. 4) being thus determined, we may now proceed to trace the minute variations of the six parts of triangles. In order to this, the general method consists in determining the relation of any two differentials. To determine this we must differentiate the formula which expresses the general relation of the quantities under consideration: the rule is very simple and well known to all who have studied the -modern analysis. Let the formula be x= ayz + b, a and b being constant quantities. Instead of x in the first member put dr. In the second, put dy instead of y, and you will have Sy.az. As there is a second variable quantity, put also dz for have dz.ay. and you 1 a and b, being constant quantities, furnish no va• riation. Hence results the differential equation, Sx=dy.az + 6%. ay; also = az; =ay. 8% If the values of dy and z are known, we thence know dx=dy. az, and "dz.ay: so that the total variation of x is constituted of two parts, the one depending on dy, the other on dz. 15. In the solutions of problems relating to the variations of triangles, we have only to substitute one by one the differentials of the sine, cosine, &c. in the appropriate formula of the problem, and proceed agreeably to the above directions. Thus, suppose that in the spherical triangle whose angles are A, B, C, and sides respectively opposite a, b, c, the sides b and c were constant. Then, from equa. (2) chap. vi. we have cos a = cos a sin b sin c + cos è cos c. This becomes by differentiation, da sin a= A sin a sin b sin c.. The formula has only two terms, because it contains no more than two variable quantities. It results from it that 16. Upon these principles a complete series of differential equations for all possible variations of spherical and plane triangles may be investigated. But as the inquiry would occupy more space than can be devoted to it in this work, it must be omitted. We shall here, however, lay before the reader, with a few alterations, corrections, and additions, the valuable summary of trigonometrical differential equations given by Delambre in his Astronomie Théorique et Pratique. OBLIQUE ANGLED SPHERICAL TRIANGLES. I. The sides b and c constant. |