298 Mr. S. P. Langley on the Amount 



In other words, it is universally true that when the numbers 

 are positive, and a, b, c, d, proper fractions, 



± a n+l + Bfln + l + Q c n+l + J) d n + 1 + < 



Aa n + hb n +Cc n + Y)d n + 



Aa n + 2 + Bb n+2 + Cc n + 2 + Bd n+2 + g 



Aa"+ 1 +B^ +1 + Cc n+1 + D^+ 1 + ; 



and hence universally true, that when the separate coefficients 

 of transmission are positive and less than unity (as is the 

 case in nature), the general coefficient of transmission in the 

 customary exponential formula is — 



(1) Never a constant, and (as determined from the custo- 

 mary formula), 



(2) always too large ; 



(3) always larger and larger as we approach the horizon. 



(4) The original light or heat of the heavenly body as 

 found by the photometric and actinometric processes, and the 

 formulae in universal use, is always too small, a conclusion 

 which we have just reached by another method. 



The above demonstration does not tell us in how great a 

 degree this coefficient is too large, and for aught we have here 

 yet demonstrated the error may be practically negligible. 



Since the method ordinarily employed demonstrably gives 

 too small results, the burden of proof might seem to rest on 

 those who still employ it, who might now with propriety be 

 asked to show that the continued use of methods and formulas 

 certainly in some degree inaccurate does not lead to an error 

 at least as great as the total absorption in question. This has 

 never been done. There is a common assumption that if 

 there were any considerable error, its results would become 

 apparent in such numerous observations as have been made 

 all over the world in stellar photometry and solar actinometry 

 during this century, since in these observations of stellar 

 magnitudes, for instance, two stars whose relative magnitudes 

 are positively known give results agreeing with the ordinary 

 formula when one is near the zenith afid the other near the 

 horizon. At first this looks almost like evidence that there 

 can be no great error in the determination of absolute mamii- 

 tudes by the ordinary formula, and yet this apparent proof is 

 demonstrably a fallacy. It is certainly a specious one, but it 

 is absolutely demonstrable that the error miglit be enormous 

 — that the actual absorption might be, for instance, 50 per 

 cent, instead of 20 — without this gross discrepancy being 

 detected by our present modes of observation. As the pre- 

 sent methods are known to give, as I have just said, values 



