Astronomical and Nautical Collections. 141 



Z'"=Y"Z=\5Y,Z 3 +18Y l YZ*+Y 3 Z 



y"'=2X"Z+ Z'" 1= (90 Y 3 Z 3 + 162 Y 2 YZ*+ 66 y, 2 Z*+42 y, y*Z) 



+(i5y J z 3 +45y 2 yz2)+(isy 1 2 z j +36y 1 y 5 z 

 + i8y„yz 2 ) +^y,y 2 z 



= 105 Y 3 Z' + 225 y 2 YZ- + 84 y, a Z* + 8 1 y, y *z 



+ y* 



tT'ss.t m Z. 



If we now select, from these general values of the coefficients, 

 those which are concerned in the horizontal refraction, when 



d = 0, and s s 1, we shall have, instead of ps zz __. + ..., 



+ ..., in which we must substitute the values of Z •••, de- 

 rived from the particular hypothesis respecting the constitution 

 of the atmosphere that we may choose to adopt. 



Example A. The simplest application, that can be made of 

 this series, is to put, instead of Professor Leslie's hypothesis of 



z*=y (n+z — nz-), merely z" = y, whence K ~ -~- =2:, and 



Z = - 2z - - s = 2 ■ - s; consequently dZ = 0, and the 

 mpsz mps 



series stops at the second term, assuming precisely the form 

 which has been actually employed, as an approximation for de- 

 termining the effect of a change of temperature. (Coll. VI.) 

 To inquire what would be the physical conditions, that would be 

 implied by this equation, would be to anticipate the contents of 

 a very elaborate memoir, which is probably now in the press, 

 and in which the author has deduced some very convenient and 

 elegant expressions, when considered merely in a mathematical 

 point of view, from a law of condensation which will scarcely 



