89 



6. Invariant Fiuictions. The variatiou of any function 'I> (x, y) when 

 operated upon by an infinitesimal transformation. 



U/=f p + '7q 



is given by 



^^^^m+"[U' 



If <1> (X, y) is to remain unchanged, tlien U <I> = o, and <I> (x, y) is a solution 

 of the homogeneous linear partial differential equation 



U / = f p + ?/ q = o, 



tliat is, "}> (x, y) is an integral of Lagrange's equation 



dx dy 



>I> (x, y) so determined is called an invariant for the transformation 



U / = f p + // q. 



A group of two or more independent transformations will not in general 

 have an invariant function. But when extended to include the coordinates 

 of n points, as in (A) above, an r-parameter group 



Uk /•(„) = f^\ s^k (Xi, yi) [^] + vk (Xi, yi) [^] I , k = 1, 2, . . . r, 



gives rise to 2 n — r independent functions 



(?»1 (Sj, yi, ... X„, J„), 92, ^3- ••• 02n-r' 



which are point-invariants of the group Qk f, and which are derived by integrat- 

 ing the r partial differential equations Uj fn = o, Uo fn ^ o, ... Ur/n = o. 

 After the manner here indicated the writer has calculated all the point- 

 invariants for the twenty-seven finite continuous groups of the plane as 

 classified by Lie.* The results appear in the Proceedings of the Indiana 

 Academy of Science, 1898, pp. 119-135. 



7. Differential Invariants. An infinitesimal transformation extended to 

 include the increment of y^ leaves invariant two functions 9^ (x, y, y^), 

 <j>2 (x, y, y'), the solutions of 



uv = f p + '/ q + '/' [^/j = o- 



The functions (p^, ^2 are called differential invariants of the infinitesimal 

 transformation U'/. Lie shows that when two independent differential 



'See Lie-Scheifers, Contin. Gruppen, pp. 360-362. 



