Instead of x in the first member put δυ. In the second, put dy instead of y, and you will have dy.az. As there is a second variable quantity, put also dz for 2, and you have dz.ay. a and b, being constant quantities, furnish no variation. Hence results the differential equation, If the values of dy and z are known, we thence know Sx=dy. az, and ♪x" 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 b cos c. This becomes by differentiation, Ja sin a=- JA 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. |