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Instead of r in the first member put or. * In the second, put oy instead of y, and you will have • d2%. there is a second variable quantity, put also 92 for 2, and you have 92. ay. a and b, being constant quantities, furnish no variation. Hence results the differential equation, da = 9/. az + 32. ay;

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If the values of 3/ and 32 are known, we thence know * = 3). az, and 94° = }z. ay: so that the total variation of r is constituted of two parts, the one depending on 3y, the other on 92. 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, — 3a sin a = – 9A 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

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16. Upon these principles a complete series of dif. ferential 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 ;". differential equations given by Delambre in his Astronomie Théorique et Pratique.

OBLIQUE ANGLED SPHERICAL TRIANGLEs.
I. The sides b and c constant.

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*c. - sin c cos b _ sin c cos b

N

3a T Tsin a T T Tsin. A K

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RIGHT ANGLED SPHERIcAL TRIANGLEs.
I. The sides b and c constant.

This case needs no formulae: for two constant sides with a right angle render the whole constant.

II. Angle B = 90°, angle c constant.

3A
a

= sin c =
Cot a

cot A

= cot A tan a

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II. Two angles B and c constant. Here there are no formulae, because there are necessarily three constant quantities in the complementary

rectangled triangle. " " a

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