As

of the complementary angle need not be treated separately. the secondary ratios vary incongruently with the angle, their differences will be negative when the difference in the angle is positive. Thus approximately

Precisely similar limits of accuracy attach to these equations as to those of the primary ratios; if we substitute 0 for 90°, greater angle for less angle, and so on.

576. To find the difference in the log sines of two angles in terms of the difference of the angles.

Here we have

Now

log sin (a + 8) - log sin a = log (cos 8+ cot a sin 8).

cos 81 and sin d < 8.

Hence the difference of the log sines is <log (1 + 8 cot a)

<μ (8 cot a - 182 cot2 a + ...) if 8 cot a < 1.

This series has terms alternate in sign and decreasing in magnitude. Hence, à fortiori, the difference in log sines is <μd cot a Again

log sin (a + 8) - log sin a = log cos 8 + log (1 + cot a tan 8). Now

cos 8>1-18 and tan d>8.

Hence the difference of the log sines is

> μ {8 cot a -18 (1 + cot2 a) +383 cot3 a - 18 (1+cota) + ...}.

Thus log sin (a + d) – log sin a differs from us cot a by a quantity which is <u8 cosec a.

This quantity decreases as a increases.

Now if

measures an angle not greater than 1', ud is not greater than 2.10-8.

Now it will be found on examining the tables that approxi

mately cosec2 a

=

5 when a is 27°; = 50 when a is 8°;

=

= 500

when a is 21°;

=

5000 when a is 48': and so increases more and

more rapidly down to 0 where it is infinite.

Hence the log sine difference formula is less and less accurate as we approach 0.

577. Hence, to interpolate a log sine corresponding to a given angle between a and a + 1', the equation

log sin (a + 8) - log sin a = μδ cot a

holds for 7 places down to 27°; for 6 places down to 8°; for 5 places down to 23°; for 4 places down to 48'; and so on.

578. Conversely, to interpolate an angle corresponding to a given log sine, in the log sine difference formula put for 8 the circular measure of 1'. Thus

log sin (a + 1') - log sin a =

μπ 10800

cot a =

N. 10-7 say.

Here N varies as cot a and therefore increases as a decreases.

Hence, as we approach zero, we find the difference in the log sines for l' is at 27° about 2478 × 10-7; at 8°, 900 × 10-6; at 21, 300 x 10-5; at 48', 90 × 10-4

Hence we can find angles, when the log sines are given correctly to 7 decimal places down to 27°, with an accuracy that increases up to of a second; when the log sines are given correctly to 6 places from 27° down to 8°, with an accuracy that increases from to of a second; when the log sines are given correctly to 5 places from 8° down to 21°, with an accuracy that increases from to of a second; when the log sines are given correctly to 4 places from 23° to 48', with an accuracy that increases from 2 to of a second: and so on.

As we approach 90°, cot a decreases; hence, we can interpolate angles from their log sines with continually less and less accuracy as the angle increases to 90°. Thus

The difference in log sines for l' is at 64° 35' about 600 × 10−7; at 81° 54' about 180 × 10-7; at 84° 34′ about 120 × 10-7; at 87° 19' about 60 × 10-7; and so on.

Thus when the log sines are given correctly to 7 decimal places, we may find the corresponding angle at 64° 35' within of a second; at 81° 54′ within of a second; at 84° 34′ within a second; at 87° 19′ within 1 second; and so on.

.. subtracting II. from I,

log tan (a + 8) - log tan a = 2μd cosec 2a....................... III.

Again, since log (1÷x) = -log x,

from III,

from II,

log cot (a + d) - log cot a = log sec (a + d) - log sec a = μs tan a from I, log cosec (a + d) — log cosec a = — Similar limits of accuracy attach to these latter equations as

- 2μ8 cosec 2a......IV.

.V.

μδ cot a

.VI.

to I.

580. We have seen that as the angle decreases to zero, the degree of accuracy with which we can interpolate from the log sine difference formula decreases rapidly. Hence it is important to overcome this difficulty. The method is explained in the following articles.

The terms in this series after the first are alternate in

sign and diminishing in magnitude. Hence the difference of the

differs from μô (cot 0 – 1) by a quantity less

Now when is small cosec2 is nearly equal to

1

; hence, 8 being less than the circular measure of 1', the above quantity is less than 10-7.

Hence a table of functions of the form log (sin 0 ÷ 0) for every minute would be one in which we could use the formula of proportional differences correctly to 7 decimal places.

It is necessary to explain how such a table could be used.

582. To explain how a table of functions of the form log (sin 0÷0) could be used.

Ө

Let contain n seconds. Then equation I. of last article may be written

L sin n" =

log

sin
0

+10+ logo + log n

II.

where a denotes the circular measure of 1" 4.848137 × 10-6.

=

By reference to the table of logarithms we shall find that log σ = 6-6855749.

Hence a table might be constructed in which the constant 4-6855749 is added to the value of log (sin ÷0) for every minute (or smaller interval) of small angles.

-

Then to find the log sine corresponding to an angle of n", we should add to this tabulated value the logarithm of n found from the table of logarithms. Moreover since n would not be more than 100000 we could interpolate for fractional values of n if required to find log n.

Conversely, to find the angle corresponding to a log sine, we should first interpolate in the ordinary tables to find an approximate value of the angle. We should then use this value to find

sin 0 log

+ 10+ logo from the tables: and subtract this from the given L sin in order to find logn by equation II. The error arising from using this approximation would be proportional to μ (cot 0-1/0), which is a very small quantity compared with μ cot 0, to which the error in log sin is proportional.

The above method is known as Delambre's Method.

583. A modification of Delambre's Method is applicable to the log sines and also to the log tangents of small angles. It is called Maskelyne's Method.

Now sin 0-0-103 and tan 0=0+303 and cos 0=1-102 approximately for small angles.

Hence the equations II. and III. may be used by employing the ordinary tables of the log cosines.

584. The method of argument in this section may be summed up as follows:

By expanding f(x + d) -f(x) in powers of d, we find that it is approximately equal to a quantity involving d. It differs, however, from this quantity approximately by a second quantity involving d2.

The limit of the term involving d assigns the limit of accuracy with which the formula of proportional differences may be used.