40°, z = 54°; the computation by means of the preceding table is as follows: 10.41311 z log. sin2 9.81592 (dL)** log. 6.9835 1.32843 D log. 2.3802 be used whenever log. a is over 2∙310. When K is less than 100,000 metres, or log. K not over 5.00, a' may be written in the 3d term for plied by if we would refer the arc du to R. Then di The results are better if the radius of curvature midway between the two latitudes is used; make dL = λ; L — λ=Lm the latitude of the middle point between L. and L', and the radius of curvature at LmRm; then should the formula (a) be multi R R dL= Rm RRm We have then = But since (See. 63 p. 103), sin' L-sin2 Lm sin (2 L) = dL sin L cos L very nearly, because a (1 — e2) e2 dL sin L cos L sin As this quantity never amounts to of dL, the third term of dr in (a) need not 1 10000 This du is the sum of the first two terms a + B. 1 See Appendix VI. p. 366. For secondary work the 4th term hк2 sin2 z.E may always be omitted. The 3d term very frequently is of no sensible value, and a1 may always be written in the place of di2, when K does not exceed 86.500 metres (54 miles), or log. K = 4.937, which comprise the vast majority of cases. When K is less than 34000 metres, two terms are sufficient. The best rule for the omission of the third term is that it need not be used unless log. a is greater than 2.31..., log. D (which scarcely varies), being about 2.38..., we shall, in that case, have log a2 D= = 7.00.. log. of 0" 001. It appears that there are hardly any cases in which the second term may be omitted. The 1st term gives the distance on the meridian of the point of departure from that point to the foot of the perpendicular from the second point, the second term gives the reduction to the parallel. It is only at a very small azimuth then, that the 2d term may be neglected even for a very short line. The 4th term may be omitted between latitude 45° and 40°, when K is not over 17000 metres, or log. K= 4.2304. Between latitude 40° and 35°, when K is not over 18500 metres, or log. K = 4.2671, and between latitude 35° and 30°, when к is less than 20,000 metres, or log. K 4.301. In computing carry log. в to 7 places, be used in making the substitution of di in (c), and the second term only when it is over 180. If in (c) we introduce di expressed in seconds obtained by (a), we must of course multiply by arc 1', and we have, finally, D-h K2 sin2 z°E, in which E may be taken out at the same time and from the same page as в and c; h could be copied from the bottom, or sum of the logs. of the 1st term, and K2 sin2 z, by taking the sum of the first two logs. of the 2d term. The formula for the difference of longitude is obtained in an obvious manner, by applying the sine proportion (Art. 81), to the spherical triangle PSS', which gives, writing du for sin dмsin P and K for sin K, L' = new latitude, computed from the formula for-dL. And the value of du in seconds of arc is obtained by converting K into seconds, by dividing K in metres by N sin 1', N being the normal, and the length of the radius used at that part of the earth in metres. The above formula thus becomes the one 1 already given (G). Lee's tables and formulæ gives a table for log. N, log. and log. (1+e2 cos2 L.) for any latitude between 20 and 50 degrees. N sin l' (G) is founded on the supposition that cos L' sin z K dм, whereas, in and if we substitute this z in the latter expression, and make it equal to 0.001", we find the corresponding log. K to be 4.4315 =log. of 27000. For any line over 27000 metres, then a correction ought to be applied to dм, or if we will allow an error of 0.002, for any line over 34000 metres about 21 miles. In the annexed table, the column headed du contains the log. of the seconds in a given arc; the column headed diff. contains the diff. between the log. of that arc and the log. of its sine (to the seventh place); the column headed K containing the log. of the length of that are in metres. To apply the correction in question after having first computed dм by the formula (G), enter the table with the given log. K, and take out the corresponding diff.; again enter the table with the computed log. dм, and take out the corresponding diff., and lastly, subtract the difference between the two quantities thus obtained from log. dм, the result will be the corrected log. dm.1 1 For denoting the difference between the log. arc and log. sin by 8, the formula (G) should be after the application of logarithms, Log. A' should be carried to eight places. Seven places of logarithms should be used for dм. For any line less than 340000 metres (21 miles) cos di omitted, being regarded as 1. may be In computing dz, sin [λ = ↓ (L+L')] is taken out to five places for main chain of triangles, and to four for the others, carrying forward dz in tenths of seconds in the first, and in whole seconds in the second. *The formula for the difference of azimuth is deduced as follows:-In the triangle rss' we have, by Napier's analogies, calling, ' the polar distance of s, s', But s' 1800 becomes cot (s'+s) = Os ( z', and s=z, and cot (s' + s) = tan [90° — (s' + s)] .'. n tandм sin(L + L') cos di (n) which is the formula (D) above, if we write dz, for tan (z' — z) and dм for tandм. The formula for dz requires some amendment within the same limits, within which we obtain di and dм; we have By transformation of (1) we get (see note p. 94, and make cos' and cos3 of dL=1) dz dм We write the second tables, into which it can every half degree of L. 0.0002. term thus, + dm3 F where log. F is to be taken from the easily be inserted, as only one value will be required for It is 7.8324 for 250, and 7.8404 for 450; diff. for 30' = The term dm3 F can never exceed 0.1. Whenever the log. of any term is not over 7.00.. the corresponding number need not be taken out. Azimuths are reckoned from south round by west, and from 0° to 360°, the signs of sin z and cos z varying accordingly. The following form, filled up with an example, is that at present used on the Coast Survey of the United States for the computation of the above formulæ for difference of latitude, longitude, and azimuth. In this form the first horizontal line expresses the azimuth of the line joining the two stations A and B; the second the angle formed by a line from A to a third station c, with AB; the third is found by the addition of these, and is the azimuth of Ac; the fourth the excess of the difference of azimuth between AC and CA over 180°, computed below at the bottom of the form; the fifth the azimuth of ca required, formed by the addition of the two above, and 1800. The sixth horizontal line contains the latitude L and longitude м of A; the seventh the difference of latitude and longitude of A |