1903.] 



The Differential Invariants of a Surface, etc. 



is 410 c.c. If the necessary corrections and estimations be made from 

 this gross cubic capacity, the weight of the brain in the Archseoceti 

 must have been considerably less than 400 grammes, and perhaps 

 nearer 300, as against that of the recent Cetacea, which ranges from 

 455 grammes in Kogia (Haswell) to 4,700 grammes in Balainoptera 

 (Guldberg). 



'"The Differential Invariants of a Surface, and their Geometric 

 Significance." By Professor A. E. Forsyth, Sc.D., F.B.S. 

 Keceived February 14,— Bead March 5, 1903. 



(Abstract.) 



The present memoir is devoted to the consideration of the differ- 

 ential invariants of a surface ; and these are defined as the functions 

 of the fundamental magnitudes of the surface and of quantities con- 

 nected with curves upon the surface which remain unchanged in value 

 through all changes of the variables of position on it. They belong 

 to the general class of Lie's differential invariants ; and some sections 

 •of them were obtained about ten years ago by Professor Zorawski, who, 

 for this purpose, developed a method originally outlined by Lie. 

 Earlier, they had formed the subject of investigations by a number of 

 geometers, among whom Beltrami and Darboux should be mentioned. 



Professor Zorawski's method is used in this memoir. In applying it, 

 a considerable simplification proves to be possible j for it appears that, 

 at a certain stage in the solution of the partial differential equations 

 characteristic of the invariance, the equations which then remain 

 unsolved can be transformed so that they become the partial differential 

 equations of the system of concomitants of a set of simultaneous 

 binary forms. The known results of the latter theory can then be 

 used to complete the solution. 



The memoir consists of two parts. In the first part, the algebraic 

 expressions of the invariants up to a certain order are explicitly 

 obtained ; in the second, their geometric significance is investigated. 



An invariant, which involves the fundamental quantities of a 

 surface E, F, G, L, M, N (these determine the surface save as to 

 position and orientation in space) and their derivatives up to order n, 

 as well as the derivatives of functions 4>, of position on the 

 surface up to order n+ 1, may itself be said to be of order n. The 

 invariants up to the second order inclusive are obtained. It appeal's 

 that, if two functions <f> and \p occur, all the invariants that occur 

 up to the second order can be expressed algebraically in terms of 



VOL. lxxi. 2 B 



