456 Analyses of Boohs. 



London tide observations for 19 years. A similar table of the Liver- 

 pool tides having been published since, he uses these results in the 

 present paper to test and improve the formulae, to which he was led 

 by the London observations. He shews, in a very satisfactory 

 manner, that the Liverpool observations have confirmed his formula?, 

 the results of the means of large masses of observation agreeing with 

 them, with a precision not far below that of other astronomical 

 phenomena, as for example, a fraction of a minute in the times, and 

 a fraction of an inch in the heights. 



Researches towards establishing a theory of the Dispersion of 

 Light, By the Rev. Baden Powell, M. A., F. R, S. 



In a paper, inserted in the last part of the Transactions, the author 

 commenced a comparison between the results of M. Cauchy's system 

 of undulations, expressing the theoretical refractive index for each of 

 the standard rays of the spectrum, and the corresponding index 

 found from observation in different media. This comparison is there 

 carried on for all the results obtained by Fraunhofer. But these 

 include only a limited range of transparent bodies ; and close as is 

 the accordance in these instances, the theory cannot be considered as 

 fully verified until we shall have extended a similar examination to 

 a greater number of media, and especially to those of higher dis- 

 persive power. The author is at present engaged in this research. 

 But has submitted a portion of his results to the public in the present 

 paper. He compares them with Rudberg's experiments, which 

 closely approximate. The substances examined are calcareous spar, 

 quartz, aragonite, and topaz. From these researches it appears, 

 that the hypothesis of undulations assigns the law and cause of dis- 

 persion in ten new cases in addition to the ten considered in his 

 former paper. 



Reseai'clies in the Integral Calculus. Part I. By H. F. 

 Talbot, Esq., F. R. S. 



The first inventors of the integral calculus observed, that only a 

 certain number of formulae were susceptible of exact integration, or 

 could be reduced to a finite number of terms involving algebraic, 

 circular, or logarithmic quantities. When the result could not be 

 attained, they were accustomed to develope the integral in an infinite 

 series. But this method is inadequate in an analytical point of view 

 to supply the place of the exact integral. Fagnani, about 1714, 

 made a great improvement. Euler further improved this branch of 

 mathematics. Abel carried the improvement still further; and the 

 author, in the present paper, details his interesting additions, and 

 the steps of the processes by which he was conducted to make them. 

 {To he continued.) 



TI. — On the Gales and Hurricanes of the Western Atlantic, 

 By W. C. Redpield, Esq., of New York. 



The object of this pamphlet is to shew, that the violent gales which 

 so frequently visit the Western Atlantic are not of an erratic and 

 abnormal character, and to demonstrate, that they are guided by 

 comparatively great regularity. The author, after a careful ex- 



