120 DR L. BECKER ON THE SOLAR SPECTRUM. 



The integration must extend from x = to x = s x — ^27(1— e' Ph ) , where Sj corresponds 



to the upper limit of the atmosphere. But since the value of the function which has to 

 be integrated is very small for the upper limit, the value of the integral will not be 

 altered if this limit be supposed to be 00. Laplace defines (loc. cit., p. 250) 



yfr(r) = e r '/e- t2 dt if T = x /r|cotg Z, 



or 



e~ r P x dx 



Jr. 



= shTzv|v^(^)- 



7cos 2 Z+2a;sin 2 Z sinZV £ 

 We arrive therefore at the result : 



F = — 



sinZ, 



Kip^ ^{.-*«>**fe±a + r ( . + j> j*].-«*3Ks±2 + ...l.. m 



mZ/r° v ^\ Jn + 1 L 7 sin 2 ZJ Jn + 2 ) ' 



and the general term within the bracket becomes 



(m-l)!L / sin 2 ZJ V^+m 



The next largest term in the series into which F (4) has been developed arises from 

 second term 

 and (6) we have 



the second term in (1 — s) 2 . At the same time — may be taken into account. By (5) 



r"0 



fi 1 , 2a 



Lsin 2 Z J 



_l=_J^_ a _ -P±-- a \e-f is + 2x 



/x (l-s) 2 sin 2 Z 



When we substitute this expression in dF (4), and write (7) 



F = ° ¥(w+l). 



sin Z ' 



the correction depending on a becomes 



{i^Z-«][^ +1 )-^+ 2 )] <»> 



C r 2a 

 sinZl 



a being less than 1 minute of arc, this correction is of no consequence. The part multi- 

 plied by 2x may be also reduced to ^ functions. We find that this term can be taken 

 into account if we replace in F (7) 



yfr(n+m) y}r{n+m)r. , „ 1 ~| , cotgZ 



v^r h y ^+^L 1 - cotg2Z+ (^^J + (^-fm)^ (9) - 



