236 
PROFESSOR FORBES ON THE EXTINCTION OF THE SOLAR RAYS 
under similar circumstances. For smaller zenith distances it has already been ob- 
served that Lambert’s simple law of the secant gives an approximation quite within 
the limits of error of the observations to be compared. I have, therefore, adopted it 
uniformly in my reductions, partly for the sake of simplicity, partly on account of a 
doubt I at one time entertained respecting the admissibility of one of Laplace’s ap- 
proximations, — a doubt which was removed by consultation with my colleague, Pro- 
fessor Kelland. I think it may be useful, however, to give a short proof of La- 
place s method, on account of its accuracy and simplicity, in which everything be- 
longing merely to the subject of refraction shall be omitted, and only what is essen- 
tial to the law of extinction considered. In doing this I shall avail myself of the 
valuable assistance afforded by Bowditch’s notes to his translation of the M^canique 
Celeste. 
34. i. To find the differential equation of the Intensity of transmitted Light. — Let 
O D. (fig. 4.) be a portion of the earth’s surface, C its centre, A' A O the path of a ray of 
Fig. 4. 
light ; C A = r the radius of any concentric stratum of the atmosphere, whose thick- 
ness A F = dr. Let the angle A' A F = E A C = v\ which denotes the angle with the 
radius made by the ray in passing through the stratum under consideration. Then 
the thickness of the stratum traversed will be A A ' = dr sec v' ; and if the density of 
the air in the stratum be §, the mass of air passed through varies as dr sec v 1 . 
35. Let e represent the intensity of the light just entering the stratum A' F, then it 
will suffer a decrement — dz in passing through the stratum A' A, bearing a constant 
proportion to the brightness of the incident beam and to the resistance which it has 
to encounter (supposing the opacity of a medium to depend solely on the number of 
