89 



6. Invariant Functions. The variation of any function <!• (x, y) when 

 operated upon by an infinitesimal transformation. 



U/=^^p-|-7?q 

 is given by 



If •^ (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 = s^ p + V q = o, 



that is, <{> (x, y) is an integral of Lagrange's equation 



dx dy 



i V 

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



U/=^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 /-(n) = isja (Xi, yi) [^] + ;;k (Xi, yO [ |^] | , k = 1, 2, . . r, 



gives rise to 2 n — r independent functions 



<Pl (Xi, y,, ... Xn, yn^ <p2, 03. ••• 02n-r' 



which are point-iwariants of the group Qk /", and which are derived by integrat- 

 ing the r partial differential equations Uj /n ^ o, U, /n =: 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 Oj (x, y, y'), 

 (f>2 (x, y, y'), the solutions of 



V'f=^V^r>o-r v' [^,] = o. 



The functions 9i, c^^ are called differential invariants of the infinitesimal 

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



*3ee Lio-Schefifera, Contia. Gruppen, pp. 360-362. 



