554 FOURTH REPORT — 1834. 



all that have passed through the element T will be found in the 

 line N H at the distance of the moon ; and going round the 

 zone, all its light in the circles described with N M, H M as 

 radii. 



The cones TAR and K L C are dark, but in I A C and 

 ICG the intensity is doubled by the section overlapping. As 

 the refracting zone is taken higher in the atmosphere H dimi- 

 nishes, and the point A approaches M till they coincide, and 

 the point M, or that point of the moon which is central in the 

 shadow, receives no light from the higher zones. 

 Putting S =s sun's semidiameter, 

 and p = moon's hor. parallax, 

 we have 



IAM = AEI + AIE = S + 2H; 



but when A coincides with M, 1 AM = p, and at the limit 



H = ^ — ^ — , or at their mean values (Baily's formulae) 



H = 20' 29"-7, 

 scarcely less than that observed by the academicians at Quito. 



The angle 



N E M = S + 2 H - y; 



M E H = S - 2W^. 



Hence, calling I the intensity of light in the pencil transmitted 

 through the element T, we obtain for the zone's addition to the 

 density of illumination in O H or N M, omitting the higher 

 powers of h, the height above the surface, 

 /n — p^ X I X d h 



integrating which betv.een the limits (H) = 35' 6" and 

 H = 20' 29"-7, we obtain half the illumination of the point M. 

 If we take H = p — S — n for the latter limit, we obtain the 

 mean illumination of a space, whose diameter is 2 h. For 

 this we require to know the relation between S and H, but it 

 depends on the law of density, of which we know little more 

 than this, that it must be between the decreasing geometrical 

 and arithmetical progressions when h increases in the latter. 

 For the present purpose it is sufficient to assume it such as will 

 represent the actual condition of the atmosphere between (H) 

 and H. This is affiarded by the observations which Gay Lussac 

 made in his celebrated aeronautic expedition. The heights as 

 given in the account of it depend on Laplace's hypothesis of the 

 decrease of temperature, being each derived from comparison 

 with the barometer of the observatory ; but we avoid this para- 



