COS a= of the complementary angle need not be treated separately. As the secondary ratios vary incongruently with the angle, their differences will be negative when the difference in the angle is positive. Thus approximately (a + 8) – cos -o sin cot (a + 8) – cot cosec (a + 8) – cosec a =- d cosec a cota. 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. - 8 cosec- a, a= + a COS < 576. To find the difference in the log sines of two angles in terms of the difference of the angles. Here we have log sin (a + 8) – log sin a = log (cos d+ cot a sin 8). Now <l and sin d < 8. 8 <p (8 cot a -48cot a + ...) if 8 cot a <l. This series has terms alternate in sign and decreasing in magnitude. Hence, à fortiori, the difference in log sines is <ud cota Again log sin (a + 8) – log sin a =log cos 8 + log (1 + cot a tan 8). Now cos d> 1-382 and tan 8 > 8. Hence the difference of the log sines is > M {8 cota - 28(1 + cot? a) + 38cot* a - 18* (1 + cot* a) + ...}. u 784 ) Thus log sin (a + 8) – log sin a differs from ud cot a by a quantity which is <}uda cosec? o.. This quantity decreases as a increases. Now if d measures an angle not greater than 1', Iuda is not greater than 2.10-8 Now it will be found on examining the tables that approximately cosecé 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 = ud cot a holds for 7 places down to 27°; for 6 places down to 8°; for 5 places down to 2jo; 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 MTT 10800 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 1' is at 27° about 2478 x 10-7; at 8°, 900 x 10-6; at 21°, 300 x 10-5; at 48', 90 x 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 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 į to ty of a second; when the log sines are given correctly to 5 places from go down to 21, with an accuracy that increases from i to šof a second; when the log sines are given correctly to 4 places from 23° to 48, with an accuracy that increases from 2 to šof 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 x 10-7; at 81° 54' about 180 x 10-7; at 84° 34' about 120 x 10-7; at 87° 19' about 60 x 10-7; and so on. Х Х Thus when the log sines are given correctly to 7 decimal places, we may find the corresponding angle at 64° 35' within is 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. a.......... 579. Since approximately log sin (a + 8) – log sin a =ud cot a ............I. .. changing a into žtt + a, log cos (a + 8) – log cos a = ud tan .........II. .. subtracting II. from I, log tan (a + 8) – log tan a = 2nd cosec 20... ... ... III. Again, since log (1 - 2) = -log x, from III, log cot (a + 8) – log cot a = – 2ud cosec 2a ......IV. from II, log sec (a + 8) – log sec a = ud tan a .V. from I, log cosec (a + 0) – log cosec a=-ud cot a VI. Similar limits of accuracy attach to these latter equations as 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. + log o .........I. sin 581. We have log sin 0 = log 0 sin (+8) sin 0 sin (+8) Now log – log = log A +8 o sin e The terms in this series after the first are alternate in sign and diminishing in magnitude. Hence the difference of the 1 Now when 0 is small cosecO is nearly equal to 72 ; hence, s 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 6 = 6) for 0 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. ......... S = 582. To explain how a table of functions of the form log (sin 0 = 0) could be used. Let contain n seconds. Then equation I. of last article may be written sin A L sin n" = log + 10 + log o + log n II. 0 where o denotes the circular measure of l" = 4.848137 x 10-6. By reference to the table of logarithms we shall find that log o = 5.6855749. Hence a table might be constructed in which the constant 4.6855749 is added to the value of log (sin 0 = 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 log + 10 + log o from the tables : and subtract this from the given L sin in order to find log n by equation II. The error arising from using this approximation would be proportional to le (cot 0 – 1/0), which is a very small quantity .compared with u cot 0, to which the error in log sin 6 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. We have sin o L sin n" = log + 10 + log o + log n ......... II. 0 tan L tann" = log + 10 + log o + log n.........III, + Now sin 0 = 0 - ZA3 and tan 0 = 0 + 383 and approximately for small angles. cos 0=1 - 282 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 (c + d)- (a) 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 d. The limit of the term involving d assigns the limit of accuracy with which the formula of proportional differences may be used. |