MOORE. — MINIMUM GEOMETRY. 219 



W G ^" H £ - -"'• 



Q dK _ H aG 



dy l dy l 





.1 



If we make the following simple transformation we can more readily 

 draw conclusions from the above equations. 



I = % x 2 = U t = v l X 12= U \ 



X X 



Suppose the transformation (T) then becomes, 



u = f(u\ v 1 ), v = g(u\ v l ). 



Then, 



"* = f (%* ^ + % (hl ) = Ul dvl > 

 and the relation (9) becomes, 



The first equation says that g is a function of v l only. Since -, is an 



u 



integrating factor and g (v) is an arbitrary function of the solution of 

 the differential equation we see from the second equation if written 



]_ 



f = u l dg, 



that f{u l , v 1 ) is the reciprocal of an integrating factor. We then have 

 for the original transformation, 



and G 2 (.x a , y l ) is the reciprocal of -an integrating factor. The trans- 

 formation then is such that, 



