1820.] Mr. Barlow's Essay on Magnetic Attractions. 301 



1 33 Correction of the Late as respects the Longitude.— It has been observed in our 

 fifth section, that, although the law for the longitude, as there determined, appears 

 to ..ive a very close approximation; yet that probably some modification might 

 he found nece'^sary in order to reduce the observed and computed results to com- 

 nlete coincidence: 1 propose, therefore, now to examine whether the principle 

 above explained will not enable us to make the correction required We have 

 seen that all the deviations produced upon the needle by the action of the ball, are 

 Dreciselv what would be found lo have taken place, by supposing the action to be 

 exerted upon some line or fluid parsing through the horizontal needle in the natural 

 direction of the dip. Now the actual deviation 

 of the inclined needle, supposing one suspend- 

 ed at C, while the ball is any where in the cir- 

 cle S E, mav be represented by ihe arc S S , 

 and this, when referred to the horizontal nee- 

 dle, would give the arc JI V. Again, 



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

 and S Z being constant for the same place, the 

 tangents of the arcs H V and S S', will have a 

 constant relation. Consequently, if the \a.^v 

 T\ hich I have deduced for the deviations when 

 the longitudeiszero, be correct, it will follow 

 also that the same law will obtain with respect 

 to the arc S S', that is, the tangent of S S' is pro- 

 uortional to the sine of the double latitude, .,00 i. 



134 Let us now suppose the ball to be placed in any other circle S B, whose 

 lon-itude is E B, the latitude being the same as in the preceding case ; then we 

 have ari-ht to assume, from what has been above stated, that the arc of deviation 

 of the inclined needle S S", in the plane of the circle S B, w-.U be the same ;^ 

 h fore viz S S" = S S' ; but still this arc, when referred to the horizon H V , 

 ^\l\ obviouOv be less than in the former instance ; and we have seen that tny expe- 

 riments seem to indicate that the tangent of this arc is to the tangent of H V very 

 nearly in the proportion of thecosine of EC B to radius, . 



\Ve uronose now to examine how far this result is approximative towards the 

 correct angle, as reduced according lo the rules of spherical trigonometry. The 

 H^i-i bein<r the arc S Z = 19° 30' (the complement of the dip), the arc S h = S S ; 

 and the aSgle Z S S" = 90° ± t (l denoting the longitude), to find the angle S Z S , 



"^Le't s'z =I'sS' or S S' = a. Z S S' = C, and the angle sought S Z S" = A. 

 Now here we have 



cut. A = 



cot. a- sin. h — cos. b. cos. C* 



tan. A = 



sin. C 

 sin. C 



— ,or 



cot. n. sin. b — cos. b. cos. C. 



Or, which is still the same, 



tan. A = 



tan. a. sin. C 



cos.ft.cos. C 



sin. b — 



cot. a. 



And this, when a is very small, is nearly equal to 



tan. a. sin. C 



tan. A = : ] 



sin. 0. 



But C = 90 ± /, therefore sin. C = cos. I, whence we have 



tan. a. cos. / 



tan. A = : r 



sin. 0. 



Now when C = 90, that is, in the case of a right angled triangle, the reduction 



of the arc S S' gives 



tan. a 



Ian. A' = -. 



sin. 



BDnnYcastte's Trigonometry, p. HI. 



