2(>2 ON REFRACTION, &C. 



d = S 5 X S. S X X racL* 



COS. M X X COS. M S X tang. Z S ^""^ ^ * = 

 «^ X tang. Z S X COS. M X x cos. M S 

 i-ad.' x S. S X 



Q. E. D. 

 €h^ke ofthe The stsi-s should be chosen so as to make the angles S 

 and X acute, as the cos. of an obtuse angle would be ne- 

 gative. 



Aclvantaps of The advantages which this method seems to possess over 

 She- Hieinod. +i, i • i i ^ . ' 



those which are already in common use, are, 1st, that only 



one observation is required, as the refraction may vary con- 

 siderably in the interval between two observations; and 2d, 

 thai: it does not require the latitude to be known, and that 

 the observation may be taken at sea with the instruments al- 

 ready in common use for lunar observations. 

 r^S'S-asfl^e! ^^'^ ^oMowmg method of solving by construction a pro- 

 Bical triangle ^lem in spherical trigonometry may possibly be new, and 

 fiventofindits worth vour insertion. 



Given the sides of the spherical triangle Z S X, to find 

 an angle Z. Let M I and M V (fig. 7) = the secants of 

 the sides Z S and ZX, including the required Z, take the 

 Z I M V zz: the remaining side; let I Z , Z V - tangents 

 of the sides Z S and ZX, and the / I Z V will be ^ the 

 required Z Z. 



JDemonst7ation. 

 Let M (fig. 6) be the centre of the sphere, join M Z, M S, 

 M X ; draw Z T and Z V tangents to Z S and Z X ; hence 

 M T and M V are thfe secants of those sides, the / TZ V 

 =: S Z X, and the Z M = the side S X. Hence supposing 

 the triangle T Z V to come into the same plane with TM V, 

 the two triangles will coincide with fig. 7. 



Q. E. D. 

 J.!'!"!m"J'^ ^^ '^ evident, that the cases of two sides and an included 

 angle being given to find the third side, and of two angles 

 and the side included to find the third angle, may be sofved 

 hy a similar construction. 



Yours, &c. 



J. B. 



,J>monstrdiion, 



the piobl 



em. 



