DR L. BECKER ON THE SOLAR SPECTRUM. 119 



under consideration and in the air at the earth's surface, Z the angle between the curve 

 of light and the radius a, then 



r/A sin v = a/j. sin Z (3). 



Combining these formulae we have 



\J cos^Z — ( 1— c - i \ + (2s— s^snrZ 



a form similar to Laplace's differential equation of astronomical refraction.* We 

 integrate this equation at first for large Z, following almost the same course as Laplace 

 has chosen for n = 0. We introduce with him the constant of refraction defined by 



2 -1 



Mo 



and consider that 



m 2 — 1 p 



Mo" 1 Po 



hence and by (2) 



l-^ = 2a(l-e-^) (5). 



Mo 



^ ^- changes from 1 at the earth's surface to 1*025 at a height of 50 miles. We may 



(1 — s ) Mo 



therefore develope (4) into a series according to powers of s. The first term becomes after 



integrating 



j, _ ,4.x -i r de-Wit* 



*o-"P (n + l)P/ I T a V 



y i yooS.Z + 2 8 in>z[ 8 -^ I <l-e-^)J 



Denote 



sin 2 L 

 and 



y = x-\ — =— or? (6). 



u sin 2 Z v ' 



Lagkange's formula for Eeversion gives 



sm 2 Z v '^ (m — l)!\sm J Z/ 



Hence the mth term of F becomes 



_ .q-i 1 r(n+m)q^|»-> -***£* f *-^*dx 

 *o,m-"Po ( m _i)!J_ s i n 2Z J e y Jco&Z + 2sm*Z.z 



* Laplace, Traite'de rn&anique celeste, Tome iv. livre x. p. 246. Paris, 1805. 



