330 
PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL IN^^ARIANTS 
complete up to a certain order, are obtained. In the second part, the geometric 
significance of the different invariants is the goal; in attaining it, some modifications 
are made in the aggregate, but they leave it algebraically complete. 
The investigation reveals new relations among the intrinsic geometric properties 
of a curve upon a surface. To the order considered, four such relations exist; and 
their explicit expressions have been constructed. 
PAKT I. 
Construction of the Invariants. 
1. In an interesting memoir'^ pulDlished in the ‘ Acta Mathematica,’ Professor 
ZoRAWSKi has developed a method, outlined by LiE,t and has applied it to the 
determination of certain properties of functions which appertain to a surface and are 
invariantive, alike under any transformation of the two independent variables and 
under any deformation of the surface that involves neither tearing nor stretching. 
In particular, he obtains the number of these functions of any order which are 
algebraically independent of one another; he also obtains expressions for several 
functions of the lowest orders belonging to recognised types. 
The method, and much of Professor Zorawski’s analysis, can be applied to obtain 
the more extensive class of all the differential functions which, appertaining to a 
surface and to any set of curves upon the surface, are invariantive under any trans¬ 
formation of the two independent varialdes. The process, which involves the solu¬ 
tion of complete Jacoliian systems of the first order and the first degree, only gives 
the invariantive functions which are algebraically independent of one another ; it is 
not adapted to the construction of the asyzygetic aggregate. Moreover, only some 
of these functions are ijivariantlve when the surface is deformed without tearing or 
stretching; they can l:)e selected liy inspection, on using the fundamental theorem 
connected with the theory of the deformation of surfaces. 
As far as possible, the notation adopted by Professor 2orawski is used. The 
analysis, preliminary to the construction of the differential equations which are 
characteristic of the invariance, is set out l)riefly; it is needed to make the process 
intelligible. There is some difference from Professor ^orawski’s analysis, mainly 
(l)ut not entirely) l)ecause a beginning is made from the consideration of relative 
invariants and not of aljsolute Invariants. 
2. The independent variables of position on the surface are taken to be x and y. 
A function jf of these varlaliles and of the derivatives of any number of functions 
* “ LTeber Bieguiigsiiivamiiten : eine Aniveiidung der Lie’schen Gruppentheorie,” ‘ Acta Math.,’vol. 16 
(1892-93), pp. 1-64. 
t ‘Math. Anil.,’ vol. 24 (1884), pp. 574, 575. 
