TRANSACTIONS OF THE SECTIONS. 531 



novelty of his theory consists, 1st, in always determining (and 

 that, in a form capable of approximate numerical computation) 

 some single value of a* {ex. gr. a^ ), which appeared not to 

 have been accomplished, for all real and imaginary values of 

 a and x ; and, Sndly, in showing that the complete formula for 

 the logarithms of a given quantity in a given base involves two 

 arbitrary independent integers, or that every quantity has an 

 infinite number of orders of logarithms in a given base, and an 

 infinite number of logarithms in each order. He suggests the 

 application of his results to the theory of numbers, of equations, 

 and of factorial functions. 



P.S. — Mr. Graves has learned, since his paper was presented 

 to the Association, that Professor Ohm, in a volume, published 

 in 1829, of his highly valuable system of Algebra, gives some 

 formulae for exponential functions which agree with the princi- 

 ples promulgated, probably about the same time, in the First 

 Part of the Phil. Trans, for 1829, but are confined to cases 

 where the given quantities are real. This distinguished Ger- 

 man analyst, however, was aware that expressions of a similar 

 kind might be obtained, which, like those of the preceding 

 Abstract, would include powers, where the root and the expo- 

 nent, and logarithms, where the number and the base, were 

 imaginary. 



Notice of the Reduction of an anomalous Fact in Hydrody- 

 namics, and of a neiv Law of the Resistance of Fluids to the 

 Motion of Floating Bodies. By John S. Russell, M.^. 



The author has been induced to contribute this paper to 

 the Transactions of the Association, in consequence of a state- 

 ment made last year by Mr. Challis in his excellent Report on 

 Hydrodynamics, the first part of which is contained in the last 

 volume of the Proceedings. The paragraph containing the 

 statement referred to is the last of the Report, and is to be 

 found in page 150, beginning " A singular fact," &c. The au- 

 thor also refers to another passage in the Report of Mr. Chal- 

 lis, consisting of the two first sentences of the paragraph im- 

 mediately before the former quotation (p. 149). 



From these statements, and many others that could readily 

 be quoted, it appears that the theoi-y of the resistance op- 

 posed by fluids to the motion of floating bodies remains in a 

 very imperfect state ; that the resistance is generally stated to 



2 m2 



