( 850 ) 
4. If we approach along the ray y= me the singular point (0,0) 
then the tangent of the integral curve is indicated by 
l vk 
1 Neree ey aes 
Fm) + «F,(m) 
So in (0,0) we have 
dy Fm) 
—_=m- - ‘ 
dx F‚(m) 
5. If the differential equation has the singular point 2 = a, y =b, 
with critical pencil, then the investigation is reduced to the preceding 
by a substitution e=a-+a, y=y+6. 
Mathematics. “On continuous vector distributions on surfaces”. By 
Dr. L. E. J. Brouwer. (Communicated by Prof. D. J. KoRTEWEG). 
(Communicated in the meeting of March 27, 1909). 
d ; 
The theorem, that a differential equation = / (« y), in which we 
7 x 
suppose f to be univalent and continuous, possesses through each 
point (z,, y,) one integral curve was proved for the first time by Caucuy *) 
for a field, in which f possesses a continuous partial differential 
quotient with regard to one of the two variables, and then by 
Lirscuitz*) for a field in which the difference quotients of f with 
regard to one of the two variables for increases of that variable 
below a certain maximum do not exceed.in absolute value a certain 
maximum. 
Prano*) finally has done away with all restrictions for f with 
the exception of its continuity, and has proved, that then still through 
any point at least one (but now in general more than one) integral 
curve exists. It is this result of Peano of which we shall make use 
to deduce a property of continuous vector distributions on a sphere 
(or on a surface equivalent to it in the sense of analysis situs, after 
it has been made measurable by a net of curves which is continuous 
one-one image of the net of principal circles of a sphere). 
1) Exerc. d’anal. 1, 1840, p. 327; comp. also Moreno, Leg. sur le calc. diff. et 
int., Tm Il (Paris 1844), leg. 26, 27, 28, 33. 
2) Bull. des sc. math. 10, 1876, p. 149. 
8) Mathem. Ann. 37, 1890, p. 482; the proof has considerably been simplified by 
Arzeta, Sull’esistenza degl’integrali nelle equazioni differenziali ordinarie, Memo- 
rie della Acc. di Bologna (5) 6, 1896, p. 33. 
