33 



On the Developemcnt of Exponential Functions ; together with several 

 new Theorems relating to finite Differences. By John Frederick W. 

 Herschel, Esq. F.R.S. Read December 14, 1815. [Phil. Trans. 

 1816, p. 25.] 



The subject here considered by Mr. Herschel relates to the cele- 

 brated theorems of Lagrange, expressing the connection between sim- 

 ple exponential indices and those of differentiation and of integration. 



Since the theorems have been demonstrated by various subsequent 

 analysts, as by Laplace, by Arbogast, and by Dr. Brinkley, the author 

 takes them for granted; but observes that in their original form they 

 are but abridged expressions of their meaning ; and that in order to 

 become practically useful, their exponential functions require further 

 development. 



And though this part of the subject has been treated with great 

 ability by Dr. Brinkley, who has deduced formulae respecting the 

 first of the two theorems far more simple than could have been ex- 

 pected from the complex nature of the subject; yet since his method, 

 when applied to the second more general theorem, would lead to de- 

 tails of extreme complexity, Mr. Herschel has taken a different view 

 of the subject ; and beginning with the more general theorem has 

 arrived at a general formula, which he believes to have been hitherto 

 wholly unnoticed, and which, when applied to certain particular 

 cases treated of by Dr. Brinkley, affords precisely the same results. 



But the mode in which this subject is treated, or even the results, 

 were not of a nature to admit of being read in public. 



On new Properties of Heat, as exhibited in its Propagation along Plates 

 of Glass. By David Brewster, LL.D. F.R.S. Land, and Edinb. 

 In a Letter addressed to the Right Hon. Sir Joseph Banks, Bart. 

 G.C.B.P.R.S. Read Januaryll, 1816. [PAi/. Trans. 1816, j9. 46.] 



In a. paper communicated to the Society in May 1814, Dr. Brew- 

 ster observed that glass, when raised to a high temperature, had the 

 property of depolarizing light, and in this respect resembled crystal- 

 lized substances ; but he did not at that time succeed in tracing a 

 resemblance in other points, which he left for future investigation. 



On resuming this inquiry in the present paper, the subject is di- 

 vided into two parts ; in the former of which he describes the trans- 

 ient effects exhibited during the propagation of heat along plates of 

 glass, whether received from adjacent bodies or communicated to 

 them ; and in the latter he describes the permanent optical proper- 

 ties produced in glass by being suddenly and partially cooled when 

 red hot. 



The phenomena here noticed depend on the total or partial de- 

 polarization of light, previously polarized by reflection at a certain 

 angle from a surface of black glass. It is well known that when a 

 ray of light thus polarized in the plane of primitive incidence is in- 

 cident upon a second surface of black glass at the same specific angle, 



VOL. II. D 



