56 



SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 68 



Similar developments may be made for F"(x), F^"'{x), . . . . , 

 whence using the undetermined coefficients C-^, Q, .... as multi- 

 pliers and adding : 



"na*(i+w^) 



F{x)+C,F"{x)+C,P-{x) + 



■■2<'^fi^) + 



4!2 



na- p 





+ !^QJrU)+ j 



Putting fei = w, k2 = n{i+n^),k^ = n{^ + ion- + 2>n*), and equating co- 

 efficients, it follows then that 



a'k,fix) =F(x)-^F"{x) + iQ V-Mi ^' pv(^) .... (I) 

 ^t24 ^ k^- 5760 ^ ^ ^ -^ 



Now letting 2ca be the portion of the energy curve covered by the 

 slit plus the bolometer, c being equal to i + w and therefore 2ca to 

 2{^a + b), then 



c^a^ 



F(x + ca) =F{x) +caF'(x) + "-—F^ix) 



3! 



4! 



Forming a similar development for F(x — ca) and adding and trans- 

 posing, there follows 



p"(..\^ 2 l F(x+ca)-\-F(x— ca) 

 ^ ^ c^a^ \ 2 



c^a* 



4! 

 which may be put in the form 



c*o^ 



Substituting in ( i ) there follows that 



a-kJ{x)=F(x)--l^^ r27^</>i(-^) 



Now 



and 



+ 5760^1^ ^^+- 



(2) 



«/.,(.i- + ca)=(/>,(.v)+c-flc/>/(A-) + '']'|%,"rx)+ ''^'l' c^^"i(.r)+. 



