(86). In addition to these we shall here put down a few other useful expressions immediately deducible from the four equations which we had occasion to investigate at p. 194; and which are as follows:

From (22) and (23) we have,

sin.2 (A+B) cos.2 c = cos.2 C cos.2 (a—b)

sin.2 (AB) sin.2 c = cos.2 C sin.2 (a - b).

Hence, by addition,

sin.2 (A — B) sin.2 c + sin.2 (A + B) cos. 2 c = cos.2 C. (30).

In like manner, from (24) and (25),

cos.2 (AB) sin.2c+cos.2 (A + B) cos.2 c = sin.21 C.. (31).

Again, from (26) and (27), we have

sin.2 (a + b) sin.2 } C = sin.2 } c cos.2 1 (A — B)

sin.2 (a - b) cos.2 1 C = sin.2 c sin.2 (A — B);

and, by addition,

sin.2 (a - b) cos.2 C + sin.2 (a + b) sin.2 1 C = sin.2 c... (32);

and, in like manner, from (28) and (29), we get

cos.2 (ab) cos.2 C + cos.2 (a + b) sin.2 1 C cos.2 c . . . (33).

CHAPTER III.

ON THE RELATIONS BETWEEN THE CORRESPONDING VARIATIONS OF THE PARTS OF A TRIANGLE.

In the present chapter we propose briefly to examine into the effect produced on the sides and angles of a triangle, by a small change taking place in the magnitude of one of them; that is, to estimate the amount of error affecting any part which may have been determined from data, not strictly accurate, and thence to ascertain under what circumstances a small inaccuracy in a proposed datum will least affect the accuracy of the result. This becomes a very essential matter of enquiry in all the more delicate practical operations of trigonometry, because, since the data furnished by observation necessarily fall short of strict accuracy, on account of the imperfection of instruments, and other unavoidable defects, we ought to know under what circumstances our observation should be made, so that the small error with which it

is affected may have the least possible influence on the quantity to be determined from it. The following problems will sufficiently show the method of arriving at this knowledge.

PROBLEM I.

In a right-angled plane triangle, whose base is b and altitude a, it is required to determine the error committed in calculating a, by means of the given base b, and the observed angle opposite to a.

Let us consider & to represent the true angle opposite to a, from which that given by observation varies by a small quantity, which we shall represent by da, and call the variation of a, then the sought side, which would be given by the equation a=b tan. a, is affected by an error du, so that instead of a it is a + da, and this we determine from the equation

adab tan. (a + da);

in which, by subtracting the preceding equation, we find the value of da to be

Now, by hypothesis, da is very small, so that we may substitute it for its sine, and cos. « instead of cos. (a + da),

in which expression da is the length of the arc to radius 1, which measures the angular error.

To determine what length b must be to render the variation da, the least possible under the same amount of error da in

we have

hence da will be the least possible when sin. 2a is the greatest possible, that is, when a=45°: so that in order to determine the height of a tower or steeple, &c. with the utmost accuracy, by means of an observation of its angular altitude, we should make the observation at a distance from the object as nearly as possible equal to its height.

PROBLEM II.

In a right-angled spherical triangle is given one of the oblique angles to determine the variation of the opposite side, arising from a small variation of the hypotenuse.

Let A be the constant angle, a its opposite side, and c the hypotenuse; then

sin. a sin. A sin. c

sin. (a+da):

sin. A sin. (c + dc)

by subtraction,

sin. (a+da) — sin. a sin. A {sin. (cdc) — sin. c};

that is, (page 58, equa. 27,)

2 cos. (a + da) sin.

... sin. da =

da = 2 sin. A cos. (c + dc) sin. dc

sin. A cos. (c+ 1⁄2 de) sin. de

cos. (a+da)

or, substituting for sin. A its value from the first equation,

which variation will be the least possible when cot. c is least, or when c = 90°. It would seem from the expression for da, that in this case da is absolutely 0, which we know cannot be. Indeed, no result deduced like that above, from a process in which certain small quantities

T

are rejected, can be considered as perfectly accurate, although they may approximate so nearly to the truth as to be practically admissible as such. If we restore the dc which has been neglected, and write the above result thus.

then, in the case of c=90°, the expression becomes

or, considering the very small arc dc to be equal to its tangent, we have in the case supposed

δα =- tan. a (dc)2,

the same expressionas other wise determined by Professor Airy in his Treatise on Trigonometry, in the Encyclopædia Metropolitana.

PROBLEM III.

In an oblique-angled spherical triangle are given two sides to determine the variation produced in the third side by a small variation of the opposite angle.

Let a, b, be the two given sides, C the included angle, and c the side opposite to it. Then

cos. ccos. a cos. b + sin. a sin. b cos. C

cos. (cdc) = cos. a cos. b + sin. a sin. bcos. (C + dC);