378 Prof. Sylvester on a Generalization of a 



powerful screen to intercept the terrestrial rays, and any agency 

 that removes them and establishes the optical continuity of the 

 atmosphere must assist the transmission of terrestrial heat*. I 

 think it may be affirmed that no sensible quantity of the obscure 

 heat of the moon, which, when she is full, probably constitutes a 

 large proportion of the total heat emitted in the direction of the 

 earth, reaches us. This heat is entirely absorbed in our atmo- 

 sphere ; and on the evening in question it was in part applied 

 to evaporate the precipitated particles, hence to augment the 

 transparency of the air round the moon, and thus to open a door 

 in that direction for the escape of heat from the face of my pile. 

 The instrument, I may remark, was furnished with a conical 

 reflector, the angular area of which was very many times that of 

 the moon itself. 



I remain, 



Yours very faithfully, 

 October 21, 1861. John Tyndall. 



XLVIII. On a Generalization of a Theorem of Cauchy on Ar- 

 rangements. By J. J. Sylvester, M.A., F.R.S., Professor of 

 Mathematics at the Royal Military Academy, Woolwich f. 



IN a paper "On the Theory of Determinants" in the Philoso- 

 phical Magazine for March in this year, Mr. Cayley has 

 referred and added to a theorem of Cauchy deduced from the 

 latter' s method of arrangements, viz. that if we resolve an inte- 

 ger n in every possible way into parts, to wit « parts of a, /3 parts 

 of b, ... of /, {a,b, c . . .1 being all distinct integers), then 



S \ = 1 



nua«n/3bf i ...n\i K 



Now both Cauchy's theorem and Mr. Cayley's addition to it 

 (which essentially consists in the observation, that if before the 

 numerator 1 in the above quantity under the sign of sum- 

 mation we write ( — )«+/*+ ■••+^ the sum becomes zer ) are no 

 more than particular cases of the following theorem : viz. that if 



* I was going to add "into space;" but the expression might lead to 

 misapprehension. My experiments indicate that the absorption of water 

 is a molecular phenomenon. If we suppose the aqueous vapour of the 

 atmosphere to be condensed to a liquid shell enveloping the earth, the ex- 

 periments of Melloni would lead us to conclude that such a shell would 

 completely intercept the obscure terrestrial rays. And if the vapour be 

 equally energetic, our atmosphere would prevent the direct transmission of 

 the obscure heat of the earth into space. On this point, however, I wisli 

 to make some further observations. g 



t Communicated by the Author. 



