Rit (Vísindafélag Íslendinga) - 01.06.1931, Blaðsíða 36

36
It will be evident, from (47b) that x=ai++ is the limit
introduced, and at this limit we shall have:
■y “ =0 for r < n — 1
dxr
The frequency curves derived from (49a) in connection
with (49) are, for n = 1, 2 and 3 in accord with our results
given in paragraphs F. I. and K. respectively.
I shall now illustrate the introdudion of two Iimits by
an example.
The frequency curve
y = eax (C+Cisinbx+Cocosbx) (50)
may be represented by a differential equation of 3rd degree, i.e.:
^ -3a g +(3a2+b2)^ -a(a2+b2)y = 0 (51)
Multiplying (51) by x—hi we introduce the limit x=hi
provided that y=0 and ^Y = 0 for x=hi.
dx
Hence it follows that y=Ceax[l — cosb(x—hi)]
(50a)
Multiplying (51) by (x —hi) (x—h2) we introduce another
limit x=ho if cosb(x—hi) = cosb(x —h2)
or
2?t
h2 — hi
Now we put (x—hi) (x—h2) = (x—^i)2—2p (x—Xi) + p2—q2
where p = (h2+hi)2—+ and q = (h2—hi)/2,
we shall therefore start from the differential equation:
dy
[(x—A,)2 —2p(x—Ai)+p2—q2]
+y
dx3
3a dJ (3a2+
+ a(a2 + -2|y
dx
(51a)
whence the formulae (16A) to (16C) furnish us with the relations: