On Refraction. 455 



5d. Thermometer within and above 4Q°, put d = k° —49°, 



then : 



L. /,(Z _-~£r) +L.'£ + L. (IOOOO - 23d) + 



6-29303 as L. Ref." 

 3d. Thermometer without and below 45°, put d = 45° — A , 



then : 



L. /,(Z - *?!&*) + L.fl + L.(10000 + 81.4 -!' 

 6*29303 as L. Ref." 

 4thly. Thermometer without and cZw;e 45°, put d = 

 /i°-_45 , then: 



L. /,(Z - ^| -r ) + L. £ + L.(10000 - 9ld) + 



6-29303 = L. Ref." 

 But as it appears more simple to avoid the two numbers 

 49 and 45,, and reckon the state of the thermometer from 

 zero, we may reduce the whole of these equations to that 

 temperature, and then find other multipliers for the number 

 of degrees above that point ; which is easily done as fol- 

 lows .- Put R as the refraction at any given temperature; 

 r = the degrees of that temperature, § = the refraction at 

 Zero, n = the multiplier for the state R, and v = that for 

 zero : then we have 



R 4- Rrrc = R (1 + rn) = §; whence R = ~— ; but 



on the contrary, p — grv == R = f (1 -- *>}j consequently, 



c(l — rv) — — - — , from which we get v = — : then, 



^ v ; l+rn 9 c r/*+l * 



by substituting Mr. Groombridge's multipliers for w, we get 

 the new multipliers for reducing the refraction from zero 

 to any other temperature. By this we also obtain the ad- 

 vantage of having only three equations instead of four; 

 which, putting h as height of thermometer above zero, will 

 now be as follows: 



1st. From zero to 49°, within. 



5S • 1192 X 1 • 1 1 76 tan. ( Z L r ) X ■ x 



V 80 J 296 



(1 — h -002147) = R/ 



2dly. For all degrees above 49 within. 



58"-1192x P127 X^(Z-igr) X -~ X 



(1 —A -002067) = R." 



2 F 4 3d. For 



