On a new Theorem in Ike Calculus of Finite Differences. 125 



of the spectrum. A small variation in the thickness of the 

 film transmits or annihilates by interference each colour of 

 the spectrum in succession. If the waves of heat be much 

 more heterogeneous (as I have already surmised) than those 

 of light, such effects would be proportionably more sensible. 



64?. Possibly a grooved surface may be considered as pre- 

 senting a number of polished surfaces, partially detached from 

 the general surface, under small obliquities to the incident 

 rays ; and we may suppose that these rays, after separation 

 by partial reflexion and refraction, reunite with unequal re- 

 tardations, producingfrst a destructive effect upon the shorter 

 waves, and suffering the others to persevere. I have already 

 adverted to the fact, that most turbid fluids.transmit chiefly 

 .the longer luminous waves. I offer these, however, but as 

 vague conjectures upon a very obscure subject. I think that 

 experiments on the colour of media, such as those we have 

 employed, and especially of. depolished plates, might not be 

 without value in illustrating the phenomena of absorption in 

 optics. ■ 



65. In conclusion, it might perhaps be expected that I 

 should take some notice of the experiments and reasonings of 

 which M. Melloni has addressed an account to M. Arago, in 

 two letters dated the 4th and 14-th of March last, and pub- 

 lished in the Comptes Rendus for the 30th of the same month. 

 These letters were occasioned by the announcement of my 

 Researches, in the same work, for the 6th of January. The 

 present paper, founded solely upon experiments undertaken 

 and completed before the despatch of the earliest of M. Mel- 

 loni's communications, will, I think, sufficiently answer all 

 the questions which are started in his letters to M. Arago, at 

 least all those in which my experiments are concerned. 



May 12, 1840. 



XXII. On a new Theorem in the Calculus of Finite Differences, 

 with its Application to the Development of the Cosine of a 

 Multiple Arc in descending Powers of the Cosine of the Simple 

 Arc. By James Booth, M./L, Principal of and Professor 

 of Mathematics in Bristol College*. 



HPHERE is perhaps no problem in pure mathematics which 

 •*■ has given rise to more discussion than the development 

 of the cosine of a multiple arc in descending powers of the 

 cosine of the simple arc. While Euler and Lagrange, on the 

 one hand, have evolved coincident results by methods very 



* Communicated by the Author. 



