677 
the initial ordinate 7, being zero of course, and the final ordinate 
ZT, being certainly unity. 
In the common case, where the frequencies 
decrease towards both the extremities, the 
integral-curve has a course like fig. 5. But if 
the first frequency, /, is still great, so that we 
: ne dl 
are inclined to admit a discontinuity in y= —, 
C d 
the curve /(w) has the aspect of fig. 6. 
Now the extended method joins to a single 
dL 
de 
in this manner we also obtain two branches 
[=®,(«) and /= D,(v) of the integral- 
curve, satisfying the relation 
_ AG faz) dD («) 
: nl Aaron nd 
value of « two opposite values of 4 
Yo 
da 
whence 
PD, (v7) + ®, (a) = constant, 
and, «, being assumed conjugate as well with z—=—o as with 
z=-+o, the constant is unity. 
So we replace the last-given 
fins 
to the 
figure by another of the shape of 
which is symmetrical with regard 
line J = }. 
The value of 7 at the point p(#=«,), 
which was formerly (fig. 6) represented 
by the ordinate pl, is now given by the 
ea difference 
DP — eb, == FPL pr. 
The increase of / in the interval py was formerly equal to the 
rise of the ordinate, viz. qQ—pP = RQ. At present it is the sum 
of the increase gQ,—pP, = R,Q, and the increase pP, — qQ, = QF, 
(fig. 7); this latter corresponds to the second branch, in which the 
axis of w is assumed to be travelled along in a negative sense. 
The area, which was formerly bounded by the curve /(z), the 
axis of 2 and the final ordinate-line «= .2,, is now found again in 
the area inclosed between the two branches #, and ®, and the 
final ordinate-line «=a, So it is as if the area of the original 
figure is so far lifted up as to be symmetrically divided by the 
line [= 3. 
= 
Now we might begin with this latter operation and derive the. 
Fra, 7: 
