302 Analyses of Books. [Oct. 



"Whence Ian. A in oblique angled splierical iriangles, having (he side a very 

 small, is to tan. A' in right angled spherical Iriangles under the like conditions, uery 

 nearly as cos. lu> radius. 



135. That the side a or S S" in our case is very small, may be shown imme- 

 dialely, as follows : 



We have seen that when/ = o, thai is in the case of a right angled triangle, we 

 have 



tau. H V = tan. S S' . cosec. S Z 



tan. H V Ian. HV 



Whence tan. S S = :— 77 = -Tr-rxrrr; . 



cosec. S Z 3-02056 



Now, in no case that Ihave reported in Section V. does the arc HV exceed 65° j 

 and, consequently, in no one of those cases does S S" exceed 2° 10', and is, there- 

 fore, small in comparison with the constant arc S Z = 19^". 



130. In order to see tlie greatest actual en or w hich can arise by Ihe approximate 

 formula alluded to, we have only to introduce into our corrected formula, viz. 



Ian. a cos. I 

 tan. A = , 



cos. O COS. C 



sin. b — . 



cot. a 



The value of cot, a, taking the arc at its maximum 2°, Ihe cotangent of which is 

 28-6362; this will give us 



tan. a cos. I 



tan. A = 



cos. b cos. C 

 sin. b 



28-63, &c. 



In which expreision it is obvious, since cos. b and cos. C are each less than unitj', 

 that the fraction 



cos. b cos. C 



28-63, &c. 



can never be greater lhanl-2Sth ; and consequenlly, the value of tan. A can never 

 be increased beyond its correct value more than l-27th of Ihe same, nor diminished 

 by more than l-29(h. 



The slight aberrations, therefore, noticed in Section V. which had somewhat 

 perplexed me hefore I fell upon the preci-ding explanation, may now be consider- 

 ed as :i confirmation of Ihe accuracy of my observations, and of lli;it particular 

 view of (lie subject which has enabled me to account for such minute discrepancies. 



137. Observations relative to the General Formula. — 



D3 cos.™ ? 



tan. A = fsin. 2 A cos. I) 



A d3 ' cos.'" 8' 



In Art. 66, where I have introduced this formula, I have spoken very cautiously 

 of whatmiiiht beth.'- probable value of m, that would render i I general for all latitudes; 

 and although I afterwards ventured to estimate it at 4, I did it with considerable 

 hesitation, and have, I trust, suflicienlly apprised the reader Ihat he ought to con- 

 sider it entirely as a matter of doubt, which I did not even consider removed by 

 the approximation it was afterwards found to give towards the observations of 

 Captain Sabine, in Baffin's Bay. (Art. 88.) 



All that I have staled on this head in the articles referred to was written pre- 

 vious to my having fallen upon the explanation of the aberrations in Ihe law of the 

 longitudes contained in the preceding articles, and will, 1 trust, convey no unfa- 

 vourable impression upon Ihe mind of the reader, as to the caution with which I 

 have made my several deductions. T have now to show, Ihat notwithstanding the 

 plausible appioximation which Ihis formula was found to give in Art. 88, the 

 value of 7?i is probably erroneously determined, at least if we admit the explana- 

 nalion given in Art. 133, etseq. 



According to the ideas there advanced, it seems to follow, that the deviation of 

 the inclined needle, or line of fluid, is wholly independent of the directive power 

 nf the horizontal needle; and, consequently, that the deviation shown by the lat- 

 ter, when acted upon by the same mass of iron, similarly posited as 10 longitude 

 and latitude, will be directly as the cosecant of the arc S Z, or as the secant of the 

 dip ; or, which is still Ihe same, inversely as Ihe cosine of the dip. Instead, there- 

 fore, of m = .3, we ought, according to this doctrine, to haven; = ). It is, how- 



