192 PROCEEDINGS OF THE AMERICAN ACADEMY. 



s s 



j S^Ri- (Is = j Si R^ ■ ds, 



which does not occur as the left member of any equation. The four sets 

 of equations (81) can be represented, by virtue of (82), as the one set 



s 



(81') jS,„R^,ds = 



for any two different numbers m and k from among the even numbers 

 not greater than 2 a (including 0) and the odd numbers not greater than 

 2)8+1, excepting m = 1, A; = 2, and m = 2, /t = 1. The equations 

 (80) determine all the constant coefficients G that actually occur in (78) 

 CCi and C.2 do not occur), while equations (81) express relations that 

 must exist between the limits s and s for any observations that conform 

 to the theory. 



To determine the values of the integrals that occur in (80) and (81), 

 we consider the functions ^ and R defined by (G2) and (68). We have 



9 So,, 

 "57 



m 



= 2m'V 





m — 1 



N • V -/ 



that is, 



9s J^ 







^^•Im a a c)Sz,n+l 



(83) ^ = _ 2 m • S„„^, , ^^^ = S,, 

 ^ 9s 9s , 



for ^ m. We have also, by (36), 



m — 1 



s.„. -5/- ^^\ i ) 2-(z-i)<'-» +2/ ^^\ i - 1 j2'i/-¥f' 



-•52m-2 ^{ ^) [^ . y 2'+l (i + i)"'+l' 



