Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 20

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If we should need them we can readily derive them from
those already stated.
We shall now consider the equation
2r=j(x(l+AV<«))^cp(q) (13)
where 2 is a function of x and q, y(jc) a function of x in-
dependent of rj, and q)(r|) a function of r] that, in most
cases, also may include x. If two of the functions z, y and
<v are known, the third one can be determined, with the re-
striction that if y is the function to be determined, qp(rj) should
not contain x except as a common factor of the form (f(x))v
where f{x) is independent of tj.
If z is the unknown, the soluíion is at once given by
each of the infinite series to A7. From (13) we derive
q> = M*(l+Aij/«0)]-í^'K*>,n) (14)
The right-hand side of (14) is also a basal function from
which qj is readily obtained.
If y is the unknown we restrict our solution to the case
that q)(q) is independent of x. When we have ascertained
that this condition is fulfilled, we apply the transformation
B, We are then able to arrive at the solution as
j'=[q)(ln[í?V(l +Ainx/<o)])]_/^z(+ri) (15)
Sometimes the transformation C enables us to surmount
the difficulties arisen from the circumstance that x is im-
plied by q)(n).
Suppose that
q>(ii) =(/(*))’M'n)
where ^(q) is independent of x.
Then we have with regard to C
^(/W)MÍW)^(1+A+))]-m1)(t|) (16)
and
yimr*]=*M ln^d+AinxJ)]-^^^) (16a)
Provided that q>(iq) does not comprise x the equation (13)