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The negative value of 1.9819 cannot here obtain, because b would in that case be negative, which is impossible.
On the Investigation of Differential Equations for esti
mating the minute Variations in the Sides and Angles of Triangles.
1. By reason of the imperfection of instruments, the unavoidable though generally slight inaccuracies of observers, the effects of parallax, refraction, the precession of the equinoxes, the varying obliquity of the ecliptie (see note, p. 119), and other causes, the sides and angles of either plane or spherical triangles, whether observed on the earth or in the heavens, can never be taken with perfect exactness. It, therefore, becomes necessary in cases where great accuracy is required, to strike out some means of estimating the extent of error which may be occasioned in certain sides and angles of triangles, by any assignable or supposable errors in the other parts. This is an interesting department of research, in which the celebrated Cotes in his treatise De aestimatione errorum in mixtá mathesi, and many subsequent mathematicians have laboured with considerable ingenuity and success. 2. The inquiry before us is one (of very few in my estimation) in which the contemplation of magnitudes as augmenting and diminishing by differences leads to a more natural and satisfactory explication, than that in which magnitudes are regarded as varying in conse
quence of motion. Hence I shall in this chapter employ the notation of finite and infinitesimal differences, instead of that of flutions: although I am fully persuaded that in a great majority of mathematical inquiries the fluxional notation and metaphysics are preferable to those of differentials. 3. When variable quantities augment or diminish by portions which are finite or susceptible of mensuration, the portions which constitute the augmentation or dimimution are called differences. If the variations, instead of being finite, are indefinitely small, they are called differentials. The former are aptly denoted by the capital Greek letter A placed before the letter which represents the variable quantity, as Air, Ay, Az, &c. the latter by the small or lower case Greek letter 3, as or, *y, oz. The processes of differentiation and integration being similar to those of finding furions and fluents, as taught in our standard works”, need not here be exlained otherwise than in connexion with the present investigation. 4. To determine the difference, or finite variation of the sine of an arc or angle, let us take from formula (U), chap. iv. the equation, sin A — sin B = 2 sin à (A – B) cos; (A + B). Here, supposing A greater than B we may denote by AB, the augmentation which A must receive to become equal to B; and, in like manner, by A sin B, the differ-. ence which must be added to sin B to make it equal to sin A. Thus, we shall have A = B + AB, and sin A = sin B + A sin B. Substituting these values in the preceding equation, it becomes A sin B = 2 sin AB cos (B + AB) .... (1.) 5. To obtain the difference of the cosine, take from the same class of formulae the equation, cos B – cos A = 2 sin . (A – B) sin à (A + B);
* See the treatises on Fluxions by Maclaurin, Simpsou, and Dealtry.
and, performing an operation similar to the preceding, there will result, – A cos B = 2 sin AB sin (B + 3 AB) .... (2) 6. To find the difference of the tangent of an arc or
angle, assume -
tan A – tan B = -
- cos A CoS is
- sin A cos B — sin B cos A - cos A. CoS is sin (A – B) = cos A CoS B ; and, operating as before, there will result, Atan B = FIR . #: AB) ' ' ' ' (3.)
cing the requisite substitutions in the value of sin A — sin B, form. U, chap. iv. obtain
2 sin #2 b cos (B + 4*n)
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 (B + AB) according to form. U, chap. iv. have Asin B = 2 sin eB (cos B cos AB – sin B sin & #4B) = 2 sin #4B cos B cosłAB – 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 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 9 sin B = sin & B cos B – sin B (1 — cos 3B); which, since sin 3B = 3B, and cos on = 1, reduces to 3'sin B = • B cos B. . . . (7. 11. Proceeding similarly with equa. (2), we obtain – 3 cos B = ob sin B, or 3 cos B = — & B sin B . . . . (8.) 12. In like manner from equa. 3 and 4, we obtain
#: or 3 cot B = — . . . . (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 3 versin B = — 3 cos B = 3B sin B.... (11.)
- sin B Also, since tan B = H, we have Cosh
- l Lastly, since cosec B = Tina we have
.* = – 25 cots coseca .... (18.) 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 a = ayz + b, a and b being constant quantities.