Vol,. 6, 1920 
MATHEMATICS: A. C. LUNN 
25 
The proof here sketched, in which the commutator is to be considered 
defined directly by (1), shows that for the case of real variables it is suffi- 
cient to assume the existence of continuous, first partial derivatives only 
of the ^'s and 77's. The basis of the proof is formed by familiar theorems 
on the existence and uniqueness of the solutions of differential equations 
and their differentiability with respect to parameters. A minimum range 
of variation for which the results can be known to apply could be specified 
by the use of inequalities such as naturally occur in connection with those 
theorems. 
With the canonical parameters denoted by a and j8, and the corre- 
sponding finite transformations by and Y^, let the point (xiq, . . . .yXno) 
be transformed first by Y^ then by X^, defining thus the functions Xi{a, /3). 
These are then determined by the differential equations 
D^Xi{a,(3) = ^i[x{a,^)l (2) 
in which the values initial for a* = 0 are determined by 
D^XiiO,^) = r^^[<OM (3) 
where Xi (0, 0) is to be Xi^. The functions Xi {a, j8) then exist for a cer- 
tain range of variation of a and jS; and with respect to the latter they 
have derivatives satisfying the equations 
D^D^x,{a,^) = 2 ^ij[<a,^)]D^Xj{a,^), (4) 
J 
in which means d^i/bxj. Now DaY]i[x{a,^)] exists and is given by 
D^rjd<a,l3)] = ^ Vij[<cx,m^j[<oi,m, (5) 
j 
which by virtue of the assumed identical vanishing of {XY) is equivalent 
to 
D^rjMcc,^)] = 2 ^ij[<0i,mvj[<cx,^)l (6) 
j 
Comparison of (4) and (6) shows that the D^Xi{a,0) and the 'qi\x{a,^)] 
satisfy the same system of linear differential equations, and according 
to (3) they agree for a = 0. Hence, for every q;,/3, 
D^Xi{a,^) = m\<0L,m' (7) 
Thus the functions Xi{a,^) are the same as would be obtained by the 
opposite sequence, X^ followed by Y ^, showing the condition in question 
to be sufficient. 
A sense in which the vanishing of {XY) is also necessary appears like- 
wise in connection with the differential equations. If for both orders of 
