314 
MATHEMATICS: G. D. BIRKHOFF 
tained including in one instance an entire extra vascular cylinder in the 
pith of Ricinus. 
A fuller account accompanied by photographs and photomicrographs 
will be published in the Journal of Agricultural Research. 
[Since this was written I have also obtained small overgrowths on 
cauliflower leaves with both formaldehyde and formic acid but not 
with vapor of ethyl alcohol or of acetone. E. F, S.] 
DYNAMICAL SYSTEMS WITH TWO DEGREES OF FREEDOM 
By George D. Birkhoff 
DEPARTMENT OF MATHEMATICS. HARVARD UNIVERSITY 
Communicated by E. H. Moore, March 12, 1917 
The present note contains a brief summary of a paper with the same 
title which is about to appear in the Transactions of the American Mathe- 
matical Society. 
The equations of motion of the dynamical system under considera- 
tion are taken in the variational form due to Lagrange 
dt ~ ^« = dt ~ = 0, 
where y represent the two coordinates of the system and where L is 
a quadratic function of their time derivatives x' ,y . By an appropriate 
change of variables the principal function L is reduced to 
L = h {x'' + y^) + ax' + ^y + 7, 
and the equations of motion then take the simple form 
involving only the two arbitrary functions X, 7, of x, y. In the reversible 
case, i.e., the case when linear terms are lacking in L, we have X = 0. 
The normal form just written is known in this special case, but is new 
in the general case so far as I have been able to determine. Any con- 
formal transformation of the variables x, y joined with the correspond- 
ing transformation of t leaves this normal form unaltered. 
By means of the transformation theory thus obtained it is established 
that a necessary condition for the existence of an integral linear in x' , y' 
is that the curves X/7 = const, form an isothermal family in the x, y-plane 
When this condition is satisfied suppose the particular transformation 
of the variables x, y to be made which takes the isothermal family into 
the family y = const. ; if and only if a linear integral exists will the 
resulting functions X, 7 become functions of y alone; and in this case 
