3S 



by the property of being propagated with a constant velocity, and 

 without degradation, or change of form of any kind. The principal 

 object of the paper is to investigate the form of these waves, and 

 their velocity of propagation, to a second approximation ; the height 

 of the waves being supposed small, but finite. It is shown that the 

 elevated and depressed portions of the fluid are not similar, as is the 

 case to a first approximation ; but the hollows are broad and shallow, 

 the elevations comparatively narrow and high. The velocity of pro- 

 pagation is the same as to a first approximation, and is therefore 

 independent of the height of the waves. It is remarkable that the for- 

 ward motion of the particles near the surface is not exactly compen- 

 sated by their backward motion, as is the case to a first approxima- 

 tion ; so that the fluid near the surface, in addition to its motion of 

 oscillation, is flowing with a small velocity in the direction in which, 

 the waves are propagated ; and this velocity admits of expression in 

 terms of the length and height of the waves. The knowledge of 

 this circumstance may be of some use in leading to a more correct 

 estimate of the allowance to be made for leeway in the case of a ship 

 at sea. The author has proceeded to a third approximation in the 

 case in which the dejjth of the fluid is very great, and finds that the 

 velocity of propagation is increased by a small quantity, which bears 

 to the whole a ratio depending on the square of the ratio of the 

 height of the waves to their length. 



In the concluding part of the paper is given the velocity of pro- 

 pagation of a series of waves propagated along the common surface 

 of two fluids, of which the upper is bounded by a horizontal rigid 

 plane. There is also given the velocity of propagation of the above 

 series, as well as that of the series propagated along the upper sur- 

 face of the upper fluid, in the case in which the upper surface is free. 

 In these investigations the squares of small quantities are omitted. 



March 15, 1847. 



Contributions towards a System of Symbolical Geometry and 

 Mechanics. By the Rev. M. O'Brien. 



The distinction which has been made by an eminent authority in 

 mathematics between arithmetical and symbolical algebra, may be 

 extended to most of the sciences which call in the aid of algebra. 

 Thus we may distinguish between symbolical geometry and arithme- 

 tical geometry, symbolical mechanics and arithmetical mechanics. This 

 distinction does not imply that in one division numbers only are 

 used, and in the other symbols, for symbols are equally used in both ; 

 but it relates to the degree of generality of the symbolization. In 

 the arithmetical science, the symbols have a purely numerical signi- 

 fication ; but in the symbolical they represent, not only abstract 

 quantity, but also all the circumstances which, as it is expressed, 

 affect quantity. The arithmetical science is in fact the first step of 

 generalization, the symbolical is the complete generalization. 



