350 Astronomical and Nautical Collections. 



spherical elasticity, which is here made = yl^, the earth's radius 



being unity. P. 105]. Hence— 1-, the element of the angle 



QB 



EGH is represented by "" ^^ — ^ , or by 



2cxx(l + yy-^{ji?-l^ b-) 



i+y 



aahvi 



The angle FGB will therefore be given when the fluent of this 

 expression is found by the inverse method of fluxions. But the 

 angle GBS is given from the value of the perpendicular SQ, and 

 SDG is a right angle ; hence the angle DSB will be given. So 

 that from the given distance SB, (= x), the density as B, (= y), 

 and the angle SAD, [the apparent zenith distance,] the line SB 

 will be given in position, and consequently the point B will be 

 given, and the figure of the whole refracted ray ABC will be deter- 

 mined. Which was to be found. 



But the fluent of — is incapable of 



being expressed in finite terms. We must therefore find a series 

 in order to compute the atmospherical refraction for astronomical 

 purposes. Now, in order to reduce the fluxion to the simplest 



possible form, we may write — for x, and it will become 



z 



— hyz ___ — yzz 



\\-\-d zz \\-\-d a 

 that is, netjlecting the sign, ^£f , y being 



yz , . —aayx . —auj: -[ 

 also Ei — , [smce y — , and z = . But at the 



C '■ '' CXX ' XX -i 



