^64 Mr Galbraith on Tngonomeirical 



This applied with its proper sign to the height of the first place j ■will 

 give the height of the second above the level of the sea. 

 ' 11. Since the difference of latitude, the difference of longitude, and 

 the convergence of the meridians, are small arcs, seldom exceeding a fcvir 

 minutes, the logs, of the lengths of these arcs, measuring them in seconds, 

 may be safely employed, corrected, if thought necessary, by Table III. 

 By nautical men, possessing the ordinary smaller classes of tables in 

 which there are proportional logarithms, these may be very conveniently 

 employed, and the results will be sufficiently accurate for almost the 

 nicest purposes. 



12. In this case we have the following formulae : — 



P. L. M=log. cos A. + P.L.J9" (8) 



P.L. c=:log.coseci (^+0 + P-L.w (9) 



In words, the proportional logarithm of the difference of longitude is 

 equal to the sum of log. cos x, and the prop. log. of the perpendicular 

 arc ; and the prop. log. of the convergence of the meridians is equal to 

 the sum of the log. cosecant of the middle latitude, and the proportional 

 log. of the difference of longitude. When a table for reducing x to lis 

 employed along with this method, the calculations will be thus rendered 

 remarkably simple. 



Explanation and Use of the Tables. 



Table I. This table contains the logarithmic values of M, P, and 

 for the latitudes from 0° to 90° inclusive, to every 10° of latitude, and to 

 every 10° of azimuth. To the intermediate arguments to the nearest mi- 

 nute, wliich in all ordinary cases will be sufficiently accurate, they must 

 be found by interpolation. For this purpose differences must be taken, 

 and proportional parts for intermediate degrees and minutes found by 

 proportion. 



Table IV. is the same as Table I. expanded between the latitudes 50° 

 and 60° to every 10' of latitude, and to every 10° of azimuth, which, by 

 aid of the mean difference for every 10' of latitude for M and P afford 

 the means of getting these numbers to every minute of argument readily. 

 To this. Table V. is also similar, but adapted to the spheroid employed 

 in the trigonometrical survey, having differences to 1' of latitude in M 

 and P in the right hand side column, with differences for 10° in O at the 

 bottom to get proportional parts readily, as in the following examples. 



Required the log. M for latitude 51° 13'.5, the log. P for latitude 50^ 

 68'.3, and the log. O for latitude 6Q° 4'.5 and azimuth, or a. 73° 13 } 



1. To latitude 51° 10' (Table IV.) log. M is 7.994073 



Proportional part of difference to 8'4 is 8^ X — 1.2 rr . . — 4 



Log. M to latitude 51° 13'.6 is , . . . 7.994069 % 



