Rit (Vísindafélag Íslendinga) - 01.06.1931, Blaðsíða 32

32
B “h 2};i+12+73 +20+74+ 6(52+60+ð4 — 0 (16B)
C + 2};o+10A2};2+20A373+(78+2 + 30+)74 + 6+
+5412++120++ = 0 (16C)
D + 6^27i+18^372+(54+2 +30+)73+(368+^3+44+)74
+6++42+<52+l 144++(45042+210+)+ = 0 (16D)
E + 12^3/i+(24^.22 + 28+)};2+(252+A3+42^5)3;3
+(336A23+648+++480A32+56+)74+ 24+(5i+964+
+(288+s+204+)<53+(2400+++336+)+ = 0 (16E)
where A, B, C, D, and E substitute the terms already given
in (16a), (16b), (16c), (16d), and (16e) respectively.
Before we illustrate by examples the procedure of solving
the problems of frequency curves by the method described
just now, I desire to point out that in the examples we have
already treated for n = 1 we have supposed that öi or a
of higher index was different from zero.
This assumption was reasonable, because the frequency
curve cannot have a maximum for a definite value of x un-
less this condition is satisfied. For frequency curves with no
maximum or mode we have been obliged to introduce limits
for the integration by multiplying the differential equation
with x—c, or (x—Ci) (x—c2) where x=c or x=Ci and x=c2
are the limits introduced. For differential equations of higher
degree the discussion of maxima is more complicated.
Therefore I prefer to consider this matter from a different
point of view.
At first I wish to call attention to the interchange of x
and rj, and y and & which I have made in (41) as com-
pared with the formulae (31) to (40). Henceforth we shall
use x and y as the variables of the Laplacian transform, in
that y represents the frequency while x is the independent
variable. By these denotations & and y are the variables of
the primary or original differential equation, & being the
Thiele function and ?; the independent variable. We shall
now study the effect of multiplying the Laplacian transform
by (x — h)*.