Rit (Vísindafélag Íslendinga) - 01.06.1931, Side 38

38
Supposing that $ is a Thiele functiori, we have:
d# ,, , + 2
dí = ('*’’+ 2”
^ »■ +...) # = »
3! 7 ^
whence it follows that
^\P‘ /dV\P2 /dVips
r'(dr;j (d»/2j (d+j
p,! p2! p3! ... (2!)P2 (3!)P3...
where pi+2p2+3p3+4p4+... = r.
Further
d++)
d»;r
r! s! r/s p dr_pi?
p! r—p! s—p! d >f~p
(55)
(55a)
These relations enable us to eliminate & from (54) and,
the resulting equation, (being an identity), supplies us with
sufficient relations from which we can determine the para-
meters a0l ah a2 ... /?0, Pu + ... J'o, Yi, /2 ... +, <5,, <52, + ...
We observe further that the relations are linear with re-
gard to the quantities a0, a, ... /50, + ... y0, y, ... <50, <5, ...
A superficial examination teaches us that the equations
got in this way are identical with those previously found,
but the procedure presented here is evidently more off hand
and straightforward.
VI.
We shall now study another transformation of Thiele’s
fundion. Let us suppose that:
ta=pa2, A3=pa3, +=pa4 ... etc. (56)
Then the differential equation of Thiele’s function may be given by:
d+ +
pa (1 +a?/+
(a</)2 , (ay)s
2!
3!
+ ...) +
whence
^ + (pa—+)# = paea"+
d+
d»/
+ (pa — h) 9) = pai5
(56a)