122 



Prof. Cayley on Abstract Geometry, 



[Dec. 16, 



Du Mouvement Politique en Prance depuis 1789 jusqu'a nos jours. 8vo. 



Toulon 1869. The Author. 



The Mortality Experience of Life Assurance Companies, collected by the 



Institute of Actuaries. 8vo. London 1869. The Institute. 



The New System of Astronomy ; or, is the Earth a Pixed Star or Planet ? 



8vo. London 1869. The Author. 



The True Theory of the Earth, and Philosophy of the Predicted End. 8vo. 



Edinburgh 1869. The Author. 



The following communications were read : — 



I. " Researches into the Constitution of the Opium Bases. — Part III. 



On the Action of Hydrochloric Acid on Codeia." By Augustus 

 Matthiessen, F.R.S., Lecturer on Chemistry in St. Bartho- 

 lomew's Hospital, and C. R. A. Wright, B.Sc Received July 

 23, 1869. (See p. 83.) 



II. " On the Thermodynamic Theory of Waves of Finite Longitudinal 



Disturbance and Supplement. By W.J. Macquorn Rankine, 

 C.E.,LL.D.,F.R.SS.Lond.&Edinb. Received August 13, 1869. 

 (See p. 80.) 



III. " On Abstract Geometry." By Professor Cayley. Received 

 October 14, 1869. 



(Abstract.) 



I submit to the Society the present exposition of some of the elementary 

 principles of an Abstract m-dimensional geometry. The science presents 

 itself in two ways, — as a legitimate extension of the ordinary two- and three- 

 dimensional geometries ; and as a need in these geometries and in analysis 

 generally. In fact whenever we are concerned with quantities connected 

 together in any manner, and which are, or are considered as variable or 

 determinable, then the nature of the relation between the quantities is fre- 

 quently rendered more intelligible by regarding tbem (if only two or three 

 in number) as the coordinates of a point in a plane or in space ; for more 

 than three quantities there is, from the greater complexity of the case, the 

 greater need of such a representation ; but this can only be obtained by 

 means of the notion of a space of the proper dimensionality ; and to use 

 such representation, we require the geometry of such space. An impor- 

 tant instance in plane geometry has actually presented itself in the ques- 

 tion of the determination of the curves which satisfy given conditions : the 

 conditions imply relations between the coefficients in the equation of the 

 curve ; and for the better understanding of these relations it was expedient 

 to consider the coefficients as the coordinates of a point in a space of the 

 proper dimensionality. 



A fundamental notion in the general theory presents itself, slightly in 



