( 537 ) 
In this way however we 
do not get a continuous 
curve as solution. In general 
the function will have two 
values at 2), 2, #3 etc, and 
it will then consist of two 
branches, which have no 
#5 connection with each other. 
We have viz. not taken care, that the value yg = yy (#2) with which 
the portion of the curve between zj and 29 ends, is the same as 
y'9 = Wales) with which the portion between zy and 23 begins. It 
is this last value for y, which we have to substitute for y' in 
P(y',y, #,2)=0 and Fy (y',y, 2’, 2) = 0, if we want to find out whether 
the data are fulfilled by taking the points (2), y)) and (2, y2) as 
points (2, y) and (#, y’). 
By putting: 
F 
iss Hy, (By Cys Coy ee ow Cn =) 
in which ej, ¢g,-.. + x1 represent constants, which are still at 
our disposal, we find for wo, ws etc. also functions, which contain 
these same constants. 
If we put 
YoYo, Ys=y's- ++ n= yn 
we may solve the c’s by means of these »—1 equations, By 
substituting the values of these c’s obtained in this way in 
y=... Cr) we get between zj and any: a curve as solu- 
tion, whose coordinates do no longer show any discontinuity. In 
ed : : 
general however the quantity en will vary discontinuously at the 
u 
points Lg @ tee Ln . 
We must state here, that it is not necessary that 
v7 x Ki) <a vy <a Ly, etc. 
If this inequality is not satisfied, y will show several values at 
a given value of «, even if y, is a one-valued function. Compli- 
cations which may present themselves, e.g. that #' becomes imaginary 
for certain values of «, or that «== or similar cases, are left out 
of account here. 
S 3. Besides these general solutions, others are also possible, in 
od 
Proceedings Koyal Acad. Amsterdam. Vol. IL, 
