1100 



1QATION NAUTICAL ASTRONOMY. 



[LUX AR DISTANCES. 



the sum of two canine*, w may replace it by a product 

 from tho formula (pag<s 628), namely : 



oos.0+oo*.0-3oos.J(0+0)cos. i (0>0) 

 to that we have 



oos. a oos. a 

 cos. A cos. A' oos. (A + A') 



Subtract each side of this equation from 1, then since 

 by Trigonometry (page (">. 



1 oos. D - 2 sin. 1 i D, and 1 + cos. (A + A') 



- 2 cos. 1 i (A + A") 

 the equation after division by 2 becomes 



sin.* i D cos. * i (A + A 7 ) 

 cog. 4 {(a+<Q + d) oos. j {(a + "0 ""<) cos. Acoa A' 



cos. a cos. a' 

 - cos. * (A + A') 



] cos. A COB. A'j[ 



cos. a cos. a' cos. 1 $ (A + A') > 



Or, calling the second of these factors 1 sin. 1 0, that 

 is, co. J 0, we havo 



sin. 1 i D = cos. i (A + AO cos. 

 .. sin. J D = cos. $ (A + A') cos. 



which is known by tho name of the formula of Bonla. 

 This formula consists, we see, of two parts : tho first 

 part determines the arc 0, and tho second by means of 

 computes D. The working model is therefore this, 

 namely : 



. t {(a + <O + d} con. i {( 4- ') " <*} coi. A ro. A' 



tot. a toe. o' co. i (A +A') 

 in. { D = co. i (A + A') cot. 



III) 



J 



from which we see that the entire work for determining 

 the true distance D may be conducted by logarithms. 



It may Ixs worth while to remark, that we were fully 

 justified in assuming that the fraction in the foregoing 

 expression for sin. 2 A D could be represented by sin. 2 ; 

 for this was only assuming that the fraction must be less 

 than unity, or that the expression within the braces is 

 positive, which it necessarily is, because sin. 2 \ D is neces- 

 sarily positive. A desirable feature in this method of 

 clearing the lunar distance is, that sines and cosines are 

 tho only trigonometrical quantities employed in it ; a 

 circumstance which not only affords relief to the memory, 

 but, at the same time, facilitates reference to the tables. 



RULE FOE CLEARING THE LU.YAB DISTANCE. The 

 steps for calculating D from the formula just established 

 may Ixs described in words as follows : 



1. Add together the apparent distance and the 

 apparent altitudes'; take the difference between half 

 sum and the apparent distance, and underneath the 

 result write the true altitudes ; and, leaving a gap for 

 two lines, then write half the sum of the true altitudes : 

 the u-hoU sum may be interposed in the space thus left. 



2. These directions having been complied with, it will 

 be found that nine arcs have been written down, the ap- 

 parent distance being the first. Then, disregarding the 

 first arc, as also the fourth arc, which is the sum of the 

 preceding three, write opposite to each of tho other 

 ttven, towards the right, the word cosine, prefixing, how- 

 ever, "complement* to the first two. Refer to the 

 table of sines and cosines for the quantities thus indi- 

 cated, putting a ;>/i mark after the last cosine, and a 

 mtnus mark in front of it. 



3. Add up the first six of the numbers thus placed in 

 column, divide tho sum by 2, and subtract the cosine 

 previously marked with tho minus ; the remainder will 

 be sin. 



fThese three precepts conduct us, therefore, to the 

 value of the first of the two expressions in the formula 

 1) above : the fourth precept, following, enables us to 

 compute the second expression]. 



4. Take out coo. 0, putting a plu mark after it, and 

 it to the cosine with the + after it ; the sum will 



m. i D ; taking, therefore, the oorrepondini{ arc $ 



P, out of the tables, and multiplying it by '2, wo havo 

 tho true distance sought 



.!/)/. 



1. Suppose tho apparent distance between tho centres 

 of tho sun and moon to bo 83" 57' 33"; the apparent 

 altitude of tho moon's centre 27 34' 5"; the apparent 

 altitude of the sun's centre 48 '-'7' 3'J"; tin- true altitude 

 of tho moon's centre 28 20" 48"; and tin- true altitude 

 of tho sun's centre 48 20' 49". Required tho true 

 distance. 



Hero d - 83 57' 33", o - 27 31' o", a'- 48 27 32" 



A = 28 i-'it' -l.s', A.'- '!.-> -'(; .I!,-. 



and proceedi]];; by tho rule, tho work will stand thus : 

 d 83 57' 33" 



a 27 34 5 comp. cos. -0523300 

 a' 48 27 32 pomp. cos. '1783835 



2)159 f}9 10 



4 sum 79 59 35 cos. 9 2399686 



| sum in d 3 57 58 cos. 9 9989587 



A 28 20 48 cos. !i.!i44.V.'75 



A' 48 26 49 cos. 9'8217187 



A + 4' 76 47 37 2)39-2358900 



19 -C 179480 

 i (A + A 1 ) 38 23 48* -cos. 9-8941054 + Th(j9 two 



31 57 53$ 



. . , cos. 99285870+ 



sin. 9 7237820 



i D 41 40 27 sin. 9 8227524 = their sum. 



.'. 15=83 20 54 



In the foregoing operation more attention is paid to 

 minute quantities than is necessary in actual practice-. 

 Fractions of a second are of course always disregarded 

 at sea : seconds themselves are, indeed, very frequently 

 neglected ; but this negligence in problems like the pre- 

 sent is by no means to be commended. A table of sines 

 and cosines computed to every ten seconds such as the 

 Tables Portatives of Callet will enable the computer to 

 take account of his seconds, without entailing upon him 

 any extra work worth mentioning. Tables of logarithmic 

 sines and cosines, computed to every ten seconds, will 

 also be found in the Nautical Tables of Inman and 

 Maekay. 



The reader will have observed that, agreeably to what 

 is recommended at page 1048 of the IXTHOIHTTION-, the 

 preceding rule directs that the table employed should not 

 be applied to, till the work can be advanced no further 

 without its aid ; and that when it is once in hand, all the 

 extracts from it, previously to tho determination of (', 

 should bo made before it is laid down. As a and A are 

 always pretty close together, cos. A may be taken out of 

 the table immediately after comp. cos. o ; and as a is, in 

 like manner, in close neighbourhood to A', cos. A' may 

 be taken out immediately after comp. cos. a'. 



The above is, of course, a rigorous method of clearing 

 the lunar distance from the effects of parallax and re- 

 fraction, and provided only that the data furnished liy 

 the observations be correct, is perfectly accurate. Hy 

 help of an auxiliary table, like that which forms Table 

 XLI1., in tho second volume of Dr. .M 

 on the Longitude, tho operation may be shortened. The 

 table alluded to supplies tho value of the expression log. 

 cos. Aeon. A whi(jh k ^j^ the Lo g (irit ] tmic Dlfferaice, 

 cos. a cos. a 



and unlike some of the subsidiary tables, for abridging 

 the calculations required in the direct methodn of 

 ing the present problem, it may always In- u>d with 

 confidence; this part of the trigonometrical operation 

 may therefore be shortened without any sacrifice of 

 curacy. In general, however, the indirect methods of 

 clearing the lunar distance, though often shorter than 

 the direct methods, are proportionately deiieient in pre- 

 cision. Tho author conceives that the strir* accuracy of 



