Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/216

Rh 188 0&quot;-A&quot;~. 21211 10&quot;7 Log. r&quot; 0-1258616 Log. cos. &quot; 9-7176467 Log. sin. (0&quot;-A&quot;) -9-7264615 Log. r&quot; 0-1258616 Log. sin. &quot; + 9-9309301 Log. (r&quot;. siii. &quot;) +0 0567917 = Log. (A&quot;, sin. fi&quot;) Log. (1) -9-5699698 Log. (r&quot;. cos. A&quot;) 9-8435083 Log. cos. (0&quot;-A&quot;) -9-9275349 -97710432 -9-5699698 Log. sin. (a&quot; -A&quot;) -9 8365895 + 9 -7333803 = Log. (A&quot;, cos. Log. tan. 0&quot; + 0-3234114 No -0-5902597 AddR&quot;... ...+0-9838441 j8&quot;+..+64 35 51&quot; 5 &quot; (observed).. + 64&quot; 35 27&quot; + 0-3935844 Geoc. Lat. (comp. -obs.) + 24&quot; 5 (l)-lg.- ^) = Log. (2) ......... + 9-5950379 - tan. (a&quot; - A&quot;) ......... -9 &quot;9749319 a&quot; -A&quot; ......... 31639 9&quot; 6 A&quot; ......... 265 7 46&quot;-0 a&quot; ......... 22146 55&quot;-6 a&quot; (observed) ......... 221 47 16&quot; Geoc. Long. (comp. -obs.) ............. -20&quot; 4 Log. (A&quot;, sin. sin. Log. A&quot; A&quot;.. cos. &quot; Log. .. + 0-0567917 .. + 9-9558405 . &quot; 0-1009512 1-26169 ... 9-6324 ...- 1-3096 -0-9420 Diff. Long, in arc of great circle So that the errors of elements for the second observation may be expressed in transcription thus : da&quot;, cos. /8&quot;= - 8&quot; -7 d0&quot;= +24&quot;-5 These errors are not greater than may be looked for, in a computation upon the method we have adopted. We have computed the true distance of the comet from the earth at the second observation A&quot;. If the true distances at the first and third observations are desired, i ni we have A = ^, A &quot; = ^ , or, in the present case, cos. p cos. p A = 1-28437, A &quot; = 1-24574, so that the comet was slowly approaching the earth during the interval over which the observations extend. If it be preferred to compare with the observed right ascension and declination, the formulae (XVII.) have yet to be applied, the calculation, as will be seen, being very similar to that in the conversion of right ascension and declination into longitude and latitude. [The formation of the equations for the determination of p, p&quot;, and k will perhaps be found the most slippery part of the computation by the beginner, and we add therefore two or three sets of data from observation and the ephemeris, with the resulting equations, which may be verified for the sake of obtaining a better acquaintance with this part of the work. Comet 1870. (Coggia, August 28.) to. ft A Log. R Aug. 28-5356 Sept. 5-4551 19-4167 The equations are 46 7-0 -11 22-1 155 27-4 0-00411 41 45-7 - 5 19-2 163 7 8 00325 29 30-0 .+ 9 22-3 176 43 S 00165 Comet 1874. (Winneclce, April 11.) to. $ A Log. R April 12-60769 23-58796 May 6-47931 320 30-8 312 57-1 281 42-8 + 8 56-0 + 23 40-7 + 53 59 2 22 59-9 0-00144 33 43 2 0-00273 46 13 2 0.00412 The equations are r 2 = 1-00666- [9-96708 p + [0 01060&amp;gt; 2 k 2 = 0-16416- [972470] p + [9-S3437&amp;gt; 2 Comet 1874. t a The Great Comet of Coggia. + 46 35 3 + 45 52 + 45 28 23 27 40 3 38 22 2 49 2 16 Log. K 0-0020201 0-0032594 0-0044190 April 17-38074 93 30 1 28-37122 92 42 7 May 9-39543 92 55 24 The equations are / 2 = l-009347-[9-9151992]p + [0-3257222]p 2 r &quot; 2 = l -020559 -[0-1391891]p +[0-26020]0]-p /2 k 2 = 0- 139596 + [8 -5723285]p + [8 -0795842] p 2 It will be remarked that the apparent motion of this comet was very slow during the interval we have taken ; it afforded a case where the orbit could only be improved by increased length of observation.] We may correct the elements thus obtained for the main effect of the error due to the assumption made on commencing our calculation, by the following process, also suggested by Olbers, and applicable to the same observa tions. There are already found r, r&quot;, v, v&quot;, and p. r&quot;. sin. (v &quot;-v&quot;) (t &quot;-t&quot; R &quot;. sin. (A &quot; -A&quot;) ut *&quot;V.sin. (&quot;- )-^TT?) and ?=R .sin. (A&quot; -A &quot;) _ V&quot; ~ r &amp;gt; m. tan.*r_ ag in the ca i cu i at i on O f ^ (t&quot; -t } sm. (a - A ) ratio of the curtate distances ^- or M. P Then compute N from R . siu. (A.&quot;-A )(q -p) . m ~ p (m. sin. (A&quot; - a ) - tan. /} (t&quot; -t ) (t &quot;-t&quot;) Multiply those terms in the equations for p &quot; 2 and k- which contain M by H, and the term in the equation for p &quot; 2 which contains M s by H 2 ; the equation for p 2 not containing M or M 2 remains un changed. &quot;With this new system of equations we find corrected values of p and of r and r &quot;, and the elements of the orbit there from as before, p&quot; is now M. H. p. To apply the above formulas to our preceding example we have . 98 59 43 . 104 52 26 . 109 34 45 v&quot; -v ... v &quot;-v&quot;... A&quot; -A ... A &quot; -A&quot;... 5 52 43 4 42 1? 9 34 57 9 44 2?