366 



a and ,r ; and secondly, of x in terms of a and y ; the corresponding 

 developeraents hitherto given being incomplete. He considers the 

 principles employed in this inquiry as presenting a solution of many 

 difficulties, and illustrating peculiarities appertaining to the theory of 

 logarithms of negative quantities ; and when applied to geometry, as 

 furnishing the means of tracing the form and developing the proper- 

 ties of curves whose equations involve exponential quantities. He 

 also states that by their means various differential and other formulae 

 usually exhibited in treatises on logarithms may be rendered com- 

 plete. An appendix is subjoined containing several examples of 

 these applications of his principles. In the course of his investiga- 

 tions, the author endeavours to explain the remarkable anomaly which 

 frequently presents itself to the analyst, of developements, in which, 

 upon substituting a particular value for the variable in each, there is 

 no approximation to numerical identity between the several resulting 

 series, calculated to any number of terms, and the respective func- 

 tions which they ought to represent. He combats the paradoxical 

 opinion which has been advanced, that equations, which in particular 

 instances are numerically false, are yet analytically true ; and ex- 

 plains the difficulty by reverting to the limitations inherent in the 

 hypothesis upon which the developement is founded. He maintains, 

 in opposition to the opinions of Jean Bernouilli and D'Alembert, that 

 the logarithms of negative and positive numbers are not in general 

 the same ; and hence infers that negative numbers have occasionally 

 even real logarithms. The chief novelty of his system consists in 

 showing that any assigned quantity, relatively to a given base, has 

 an infinite number of orders of logarithms, and an infinite number of 

 logarithms in each order. 



On the Reflection and Decomposition r>f Light at the separating Sur- 

 faces of Media of the same and of diffei-ent refractive Powers. By 

 David Brewster, LL.D. F.R.S. L. % E. Read February 12, 1829. 

 [Phil. Trans. 1829,;?. 187.] 



When white light is incident upon a surface which separates two 

 different media, the portion that is reflected should, according to the 

 Newtonian theory of light, preserve its whiteness, provided the thick- 

 ness of either of the media exceed the eighty millionth of an inch. 

 But since the dispersive powers of bodies are different, it must follow 

 as a necessary consequence, that reflected light can never under any 

 circumstances retain perfect whiteness, although the modification it 

 experiences is not of sufficient amount to become sensible in ordinary 

 experiments. The author during his investigations of the laws of 

 polarization for light reflected at the separating surface of different 

 media, had occasion to inclose oil of cassia between two prisms of 

 flint glass, and was surprised to find that the light reflected was cf a 

 blue colour. The fact was new, but might be readily explained 

 upon the principle that although the refractive density of oil of cassia 

 greatly exceeds that of flint glass for the mean rays, yet the action 



