1 36 Mr. Moon on the Evaluation 



from which we obtain 



x = A cos (at + B), 

 y — A, cos (b t + B ; ), 



« = A /; cos (c t + B w ), 



where A A, A ; , . B B ; B /y are constants to be determined from 

 the initial circumstances of the motion of the particle. From 

 these equations, coupled with the fact which Fresnel assumes 

 in his demonstration of the axes of elasticity, viz. that the 

 change of position of the surrounding particles from the state 

 of rest does not affect the forces upon the disturbed particle, 

 we gather, that (I) without some special interposition of pro- 

 vidence directed to each individual particle, it 'would never 

 move at all, whatever might be the state, 'whether of rest or mo- 

 tion, of the other -particles around it ; and (2) that once in motion, 

 it Would vibrate for ever without the least reference to or influ- 

 ence upon the other particles. In my former paper, I said that 

 Fresnel was driven to make an assumption as to the velocity 

 of propagation, which rested only on the analogy of a case 

 most widely differing from that under consideration. I now 

 show that it is futile to talk of the velocity of propagation, 

 when on his own showing no wave whatever can be propa- 

 gated. 



I purpose, in a future paper, to consider FresnePs expres- 

 sions for the intensity of the reflected and refracted rays when 

 polarized light is incident on a surface. 



Liverpool, December 3, 1845. 



XXV. Reply to some Remarks contained in Prof. Young's re- 

 cent paper " On the Evaluation of the Sums of Neutral Se- 

 ries." By R. Moon, M.A., Fellow of Queen's College, 

 Cambridge, and of the Cambridge Philosophical Society*. 



INa paper published in this Journal some months ago, upon 

 * the symbols sin oo and cos oo , I entered upon the discus- 

 sion of the value of the series 1 — 1 + 1 — 1 +&c. continued to 

 infinity, which I then showed to be 1 or indifferently, in 

 opposition to the commonly received opinion, which would 



make it equal to — . Prof. Young appears to be partly of 



25 



my opinion in this respect, but seems to think I have made a 

 mistake in supposing this to hold in all cases; for he appears 

 to be of the opinion, that when the above series is considered 

 as the limit of the converging series 1 — a' + # 2 + &c., where x 



* Communicated by the Author. 



