24 
MATHEMATICS: A. C. LUNN Proc. N. A. S. 
of complete ionization; but in closing it may be pointed out that it ac- 
counts for the remarkable facts that so many very dissimilar chemical 
substances (for example, hydrochloric acid and potassium chloride) seem 
to be equally ionized, and that a volatile substance like hydrochloric acid 
does not have an appreciable vapor-pressure even in 1 -normal solution 
where 15% of it must be assumed to be in the un-ionized state, if the con- 
ductance-ratio is taken as a measure of ionization. It may also be men- 
tioned that it avoids the improbable conclusions as to the abnormal ac- 
tivity of the un-ionized molecules to which solubility-effects, interpreted 
under the older assumptions, lead.^^ 
1 Le,wis, Proc. Amer. Acad., 43, 1907 (259-293) ; Zs. physik. Chem., 61, 1908 (129-165). 
2 Maclnnes and Parker, /. Amer. Chem. Soc, 37, 1915 (1445-1461). 
3 Ellis, Ihid., 38, 1916 (737-762); Noyes and Ellis, Ihid., 39, 1917 (2532-2544). 
4 Jahn, Zs. physik. Chem., 33, 1900 (559-576). 
6 Harned, /. Amer. Chem. Soc, 38, 1916 (1989). 
6 Un\i2irt, Ihid., 41, 1919 (1175-1180). 
7 Lewis, Ihid., 34, 1912 (1635); Bates, Ibid., 37, 1915 (1421-1445). 
8 Noyes and Falk, Ihid., 33, 1911 (1454). 
9 Maclnnes, Ihid., 41, 1919 (1086). 
^° Noyes, "The Physical Properties of Aqueous Salt Solutions in Relation to the 
Ionic Theory," Congress of Arts and Sciences, St. Louis Exposition, 4, 1904 (317); 
^czewce, 20, 1904 (582) ; abstract, Z5. physik. Chem.., 52 (635); also Noyes, /. Amer, 
Chem. Soc, 30, 1908 (335-353). 
11 Milner, Phil. Mag., 35, 1918 (214, 354); Ghosh, Trans. Chem. Soc. {London), 
1 13, 1918 (449, 627); Bjerrum, Zs. Elektrochem., 24, 1918 (321). 
12 Bray, /. Amer. Chem. Soc, 33, 1911 (1673-1686). 
THE COMMUTATIVITY OF ONE-PARAMETER TRANSFOR- 
MATIONS IN REAL VARIABLES 
By Arthur C. Lunn 
Department op Mathematics, University oe Chicago 
Communicated by E. H. Moore, November 10, 1919 
If X = l^U^i. . -^J^. and Y = . -^n)^. be the differential 
symbols of tv^o one-parameter transformations, the knov^n condition for 
the commutativity of the finite transformations is the identical vanishing 
of the commutator {XY) = XY — YX. The proof given by Lie-Bngel, 
applying to the case of analytic functions, depends on expansions in 
powers of the canonical parameters. The computation of {XY) accord- 
ing to the original definition involves the use of the second partial de- 
rivatives of the coefficients ^, rj, but the final form, containing only first 
derivatives, is 
