If the pyramid be not a regular one, and A, B, C, &c. be the angles of inclination,

S = 360° n. 180° + A + B + C +

(5.) To find the solid angle at the vertex of a cone.

Let B be the vertical angle of the cone, then the part of the diameter of the sphere to radius unity, described about the vertex of the cone, corresponding to the spherical segment within the cone,

SECTION VII.

ON THE SMALL CORRESPONDING VARIATIONS OF THE PARTS OF A TRIANGLE.

133. To estimate the probable effect of error in observation; to reduce observations made in one situation to what they would be in a situation little distant; to take account of refraction, parallax, precession, &c., it is absolutely necessary to ascertain the effect which will be produced on one part of a triangle by the variation of another, all the rest remaining unaltered. In almost all cases expressions may be conveniently found by writing down two equations, one of which results from giving to the quantities contained in the other the variations which they are supposed to undergo, and then taking the difference. The advantage of this method consists in showing precisely the magnitude of the error made by any further simplification. The following examples will point out more clearly the meaning of what has been just observed. this chapter, A, B, C are circular arcs to radius unity.

In

and ▲A, and ▲a, be the contemporary variations of A and a ; B and y remaining unvaried; then

In almost all cases, AA, and therefore Aa is so small that it is sufficient to take only the first terms of the series on each side of this equation; in which case,

(Aa)2

sin ẞ. sin y

hence, if the magnitude either of AA, or Aa be given, the magnitude of the other variation is found.

If the philosophical problem require great accuracy, another and more correct approximate value of AA, or Aa, may be obtained by retaining the first two terms of the above series, and finding the value of ▲A in terms of ▲a, or ▲ a in terms of ▲ A, by the solution of a common quadratic equation.

135. Again, if a be the measured distance from the base of a building; the arc subtending the observed angle of

elevation, and r the required altitude; then

x = a tan 0;

let and be the contemporary variations of x and 0,

and

x + x = a tan (0+ ▲0),

..

Axa {tan (0+ 0)-tan e},

= a sec 0. sec (0+ ▲0). sin ▲ 0, (art. 101. pt.I.) and Axa (sec 0)2 ▲0; ▲ being, from the nature of the case, very small: thus the error of observation being given, the error in altitude is found.

136. PROB. To find when this error is the least.

By the supposition a is indeterminate, and

cot 0, by the preceding article,

which is least when sin 2 0 is the greatest; that is, when

which shows that the error in altitude is least when the angle of elevation is 45°; or when the height of the building is equal to the distance of the observer from its base.

.. sin (a +

.. sin (a +

▲a) = sin A. sin (y+ y), A being given;
▲a) — sin a = sin A { (sin y + y)

Δα

sin y)},

hence, cos(a + ^a). sin 4a=sinA.cos(+47). sin 7,

2

2

if m and n be the number of seconds in ▲a and Ay, respec

tively,

tan a

If y

=

mn

π

2'

care must be taken in the reduction, and it must

not be supposed, that, because sin a = sin A in this case,

where 0.000004848 is the arc of 1" corresponding to rad = 1.