296 Mr. S. P. Langley on the Amount 



which the photometer or actinometer does not discern. Ac- 

 cording to the usual hypothesis, the coefficient of transmission, 

 which is the quotient obtained by dividing the value after 

 n absorptions by that after n — 1 absorptions, or, more gene- 

 rally, that from the expression 



i 



/Value after n absorptions \n-i» 

 \ Value after m absorptions/ 



is a constant. It is in fact not a constant, as we shall prove 

 later; but we shall first show that, if we proceed upon the 

 ordinary assumption, the value obtained for the original light 

 of the star before absorption will, in this case, be too small. 

 For, if we observe by a method which discriminates between 

 the two radiations, we shall have, if we separately deduce the 

 original lights from our observation of what remains after one, 

 and, again, after two absorptions, the true sum 



A + J3 Aa 2 + Bb 2 ' 



while if we observe by the ordinary method, which makes no 

 discrimination, we shall have the erroneous equation 



A | B _ (Aa + B6)» 



which is algebraically less than the first, or correct value ; for 

 the expression 



(Aa) 2 (Bb) 3 (Aa + Bb) 2 

 Aa 2 + Bb' Aa 2 + Bb 2 

 readily reduces to the known form 



a 2 + b 2 >2ab. 



Moreover, since a 2 -\-b 2 — 2ab = (a — b) 2 ,the error increases with 

 the difference between the coefficients. 



Now, in the general case, if we suppose the original radia- 

 tion L to be composed, before absorption, of any number of 

 parts A 1? A 2 , A 3 , + . . . having respectively the coefficients of 

 absorption a l5 a 2 , a 3 , -{-..., the true value of L is given by a 

 series of fractions which may be written in the form 



whereas the value of the original energy by the customary 

 formula would be 



Z(Aa)\ 



