and, performing an operation similar to the preceding, there will result, - Acos B = 2 sin #4B sin (B + $4B) .... (2.) 6. To find the difference of the tangent of an arc or -angle, assume sin A sin B tan A - tan B = cos A - cos B _ sin A cos B - sin B cosa —=– sin (a - B). F cos’ costs? and, operating as before, there will result, sin AB cos B cos (n + AR) * * * * (3.) 7. Taking, in like manner, cos B CoS A sin (A-R) cot B — cot A = - - - = +-3 sin B sin A Sin A sill B we shall obtain for the difference of the cotangent sin AB sin a sin (n + AB) * * * (4.) 8. For the difference of the secant, take first, I l_ _ seca – sec s cos B – cosa = F - F = -FEER = (sec A — sec B) cos B cos A: whence, form. U, chap. iv. Making in this equation substitutions analogous to the preceding, we shall have 2 sin #AB sin(n + 4AB) Atan B = Aseca = ... ... (5.) - - cos is cos(* + As) 9. So, again, since sin A - sin B = , we may by introdu CoSec A - cosec is cing the requisite substitutions in the value of sin A — sin B, form. U, chap. iv. obtain 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 3B. Returning to the first equation, we shall, by oins COs (B + $4B) according to form. U, chap. iv. have Asin B = 2 sin #AB (cos B cos 34 B – sin B sin ##AB) = 2 sin AAB cosbeos #4B – 2 sin” 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 As 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 * 3'sin B = sin 38 cos B – sin B (1 — cos 3B); which, since sin ob = 3B, and cos 3B = 1, reduces to *sin B = 2E cos B.... (7. 11. Proceeding similarly with equa. (2), we obtain - * cos B = 3B sin B, or 3 cos B = — oh sin B....(8.) 12. In like manner from equa. 3 and 4, we obtain &n sing s : * * * (10.) 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 be deduced thus: Since versin B = 1 — cos B, we have *versin B = — 3 cos B = 3B 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 z = ayz + b, a and b being constant quantities, - . . . ." Instead of x in the first member put or. 9. In the second, put oy instead of y, and you will have • (12. **. there is a second variable quantity, put also oz for z, and you have oz. ay. w a and b, being constant quantities, furnish no variation. Hence results the differential equation, *r = oy.az + 32. ay; also # = asso - so Toy = az; -FIf the values of oy and 32 are known, we thence know * = ?y. az, and 3r" = *z.ay: so that the total variation of z is constituted of two parts, the one depending on oy, the other on oz. 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 cosa = cos A sin b sin c + cos bcos c. This becomes by differentiation, — oasin a = - 3A sin A sin b sin c. The formula has only two terms, because it contains *. more than two variable quantities. It results from it that .* - sin A sin b sine - sin a sin b sin c.- sin csin b sinc 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 trigonometrical differential equations given by Delambre in his Astronomie Théorique et Pratique. |