( 540 ) 
By treating wo in the same way as y,, we may deduce another 
differential equation: 
Its solution: 
y = sz (#, €) 
will represent the required function between the limits zr, and z,. 
In this way we may represent y again as the sum of integrals 
of Fourier, provided we can solve the differential equations / = 0, 
F,= 0 etc. The ordinates of the curve construed in this way will 
show no discontinuities, but the differential coefficient will be in 
L 
general discontinuous in the points zz... en. By taking in wi 
again an arbitrary number of constants, we shall be able to make the 
dy : : ; : 
— continuous in an arbitrary number of these points. 
Lv 
If we had solved a differential equation of the second order, 2 
would contain two constants, of which we may dispose in such a 
dy 
way, that both the yp and the = are continuous in zz, without our 
4 L 
being obliged to introduce artificially constants in y,. As in mechan- 
ical problems always differential equations of at least the second order 
occur, it follows that this solution fulfils the condition, that both 
the coordinates and their fluctions change continuously. The change 
of the second fluctions can be discontinuous in some points, but this 
is not in contradiction with the requirements of mechanics. 
Indeed in some problems the case may present itself that second 
fluctions, which were 0, get suddenly a finite value, even without 
our having to assume infinite forces, e.g. when the fibre, on which 
the apparatus is suspended, breaks or when a circuit is closed. Let 
us for instance imagine a condenser with a difference of potential 
V and a charge Q. The condenser is nearly closed by means of 
a wire. Before the circuit is closed the current i= and also the 
di Tigh NG : Ne lis | a ORS 9 
er PE will be zero. If we then close the circuit, ¢ will remain 
zero at the first moment; but 5 will suddenly assume the value V. 
These differential equations too have function-solutions. 
