which is directed along the lines y = a. The total angle described by 
this vector along c is zero, just as the one described by the existing 
vector along x. The. annular domain between x and c can thus be 
filled up as in the two preceding cases with such a finite continuous 
vector distribution, that the whole distribution inside x is now free 
of points zero. 
So we can formulate: 
Theorem 6. A finite continuous vector distribution with a finite 
number of points zero can be transf oimed, by modifications as small 
as one likes inside vicinities of the points zero which can be chosen 
as small as one likes, into a new finite continuous vector distribution 
which has as points zero only a finite number of radiating points , 
and a finite number of reflexion points. 
In particular those points zero about which the angle, described 
by the vector for a positive circuit, is positive, are broken up into 
radiating points; those about which this angle is negative, are broken 
up into reflexion points; whilst those for ivhich it is zero-, vanish. 
In a following communication we shall extend this theorem to 
distributions with an infinite (denumerable or continuous) number of 
points zero. 
§ 6 . 
Remarks on the tangent curves and singular points on a sphere. 
If we have on a sphere a finite continuous vector distribution 
with a finite number of singular points, then the reasonings of § 1 
lead with small modifications to: 
Theorem 7. A tangent curve to a finite continuous vector distribution 
with a finite number of singular points on a sphere is either a 
simple closed curve, or save its ends it is an arc of simple curve, of 
which the pursuing as well as the recurring branch either stops at a 
point zero, or enters into a simple closed tangent curve, or converges 
spirally to a circumference consisting of one or more simple closed 
tangent curves. 
From this ensues that also on a sphere a tangent curve cannot 
return into indefinite vicinity of one of its points, after having reached 
a finite distance from it, unless it be, to close itself in that point. 
Out of the reasoning of $ 1 we can deduce farthermore without 
difficulty that a fundamental series of closed tangent curves with 
the property that of the two domains determined by one of them, 
