Sophus Lie. 63 



where a is an arbitrary parameter, when 



Such a group possesses an infinitesimal transformation, which may be 

 represented by 



where & is arbitrary, and the infinitesimal transformation determines 

 the group. Moreover, the necessary and sufficient condition that the 

 function fl(a;, y) should admit the above group is that the function 

 should admit the infinitesimal transformation of the group, and the 

 analytical expression of the condition is 



If the function 12 involves ?/, where y' denotes dy/dx t say, it is 



&(%, y, y'\ 



the analytical expression of the condition, that it admits the same 

 group is 



Now Lie discovered that, if an equation 



admits the infinitesimal transformation just indicated, so that U'(/) = 

 then 



Xdy - Ydx 



is an exact differential save only in the trivial case X?? .- Yf = 0; so 

 that the transformation determines a factor of integrability, and thus, 

 merely after a quadrature, leads to the integral of the equation. 

 Further, the significance of the result is not thereby exhausted, for it 

 permits the construction of the differential equations of the first order 

 that admit any given finite continuous group of transformations, for 

 instance, a projective group. All that is necessary for this purpose is 

 to construct the infinitesimal transformation which determines the 

 group, and to obtain a couple of independent integrals, say u and v 9 of 

 the system 



dx _dy _ dy' 



j'-'-^-d^ ,<r 



dx dx 



