Cambridge Philosophical Society, 139 



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 xipper 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 ujjper 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. — Contributions towards a System of Symbolical Geo- 

 metry 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 arithmC' 

 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, 

 aj^ect quantity. The arithmetical science is in fact the first step of 

 generalization, the symbolical is the complete generalization. 



In this view of the case, the author has entitled his paper Contri- 

 butions towards a System of Symbolical Geometry and Mechanics. 

 The proposed geometrical system consists, first, in representing 

 curves and surfaces, not by equations, as in the Cartesian method, 

 but by single symbols ; and secondly, in using the differential notation 

 proposed in a former paper* to denote perpendicularity, and to ex- 

 press various equations and conditions. The proposed mechanical 

 system is analogous in many respects. Examples of it have already 

 been given in the paper just quoted. 



The author uses the term direction unit to denote a line of a unity 

 of length drawn in any particular direction ; and he employs the 

 symbols a j3 y to denote any three direction units at right angles to 

 each other. 



He defines the position of any point P in space by the symbol re- 

 presenting the line OP (O being the origin) in magnitude and direc- 

 tion. U X y zhe the numerical values of the coordinates of P, and 

 a ^ 7 the direction units of the coordinate axes, the expression 



xoc+y^ + zy 



represents the line OP in magnitude and direction, and therefore 

 defines the position of P. This expression he calls the symbol of the 

 point P. 



If r be the numerical magnitude, and e the direction unit of OP, 

 we have 



rs—xa.+yfi+zy: 



re is therefore another form for the symbol of the point P. 



* Read Nov. 23, 1846. 



