KiRSTiNE Smith 
37 
Hence the factor arising from the denominator of (43) is 
(2q + 2p-l){2q + 2p + S) -{2q-l) {2q+4p + 3) 
(2g- 1) (2g + l)2 ... {2q + 2p -3)>' {2q + 2p -l)P+^{2q + 2p + l)>'+^2q + 2p + {2q + 2p +5)" ... {2q + 4p+l)^{2q + 4p + r) ' 
The numerator of this equals {p + 2), 
multiplying with the factor previously found we therefore get 
p+iD 
{ .2P ...{p-l)^if'{p+l) P . 2(''+i) .{p^2) 
(2g-l)(2g+l)2... {2q + 2p-'i)i>(2q + 2p-l)>'+^2q + 2p+l)i'+^(2q + 2p + 3)v+^2q+2p+5)i' ...{2q + 4:p+iy^2q + 4p + S)' 
which is what we wanted to prove. 
(4) When the values of A and D are introduced in (42) we get 
ft 
,,5= 
(2g-l) (2(7 + 1)2 ... {2q + 2p-5y'-'^(2q + 2p-3}"{2q + 2p-l)"--^ ... {2q + 4p -7)- {2q + 4p - 5) 
+ a . 2M'^-i) X 
. 2''-2 ... {p-2)2(_p-l)}2 . 2t''-i)(»^-*) p 
(2?-!) (29 + 1)2 ... (2<? + 2j9-7)'-2(2? + 2p-5)"-M29 + 2;)-3)"-i(2(? + 2p-l)"-M25 + 2p + l)"-2 
« _ _ 9/>-2 . .. (p - 2)2 (p - 1 ) 1 2 . 2''(''-i) [1 + ap [2q + 2p -3)] 
or 
(2q + 4p-l)-(2q + 4p~5) 
(45). 
(2g-l)(2g + l)2 ... (2? + 2p - 5)"-i (2g + 2^) - 3)" (2? + 2p - l)"-i ... (2q+4p -If [2q+4p -5) 
The denominators of the formulae (38) — (41) for ^,0^ are now known since they 
12 
only consist of the factors and ^S. To be able to write down the general expression 
for „ct'^ we should have to evaluate the minors of 8, but their form is so complicated 
that a direct calculation of the determinants for the degrees of function in question 
appears to be simpler. AVith the material in hand we are however able to deter- 
mine „ct'^, for a; = 0 and = 1. 
(5) From (38) and (39) we see that 
a;=0 
pO-'f, = ajj+icrf, = ^ (1 + a), and with the 8's as given by (45) 
x=(i a; = n 
(r2(l+a)[l+ap(2p + 3)] 1 . 32 . 5' . 7^ . O'... {2p - 1)" (2ff + !)"+! (2p + 3)" (2^+5)"-^ ... {4p-l)2 {4p + l) 
N \ l .2 .3 ...p}^ . 22.'. [1 +a (p + 1) (2?) + 1)] 5 . 72 ... (2j9-l)"-2(2j5+l)"-i(2p+3)''(22)+5)"-i ... {4p - (4^9 + 1) 
ct2 (1 4- a) 32 . 52 ... (2;^; - 1)2 (2/; + 1)2 .[l + ap (2p + 3)] 
or 
iV{l .2.3...:p}2.22f .[1 + a(^+ 1)(2^+ 1)] 
^=o_ CT2j3 5 2p-l 2^9 + 1)2(1 + a) [l + ap (2^ + 3)] 
iV|2-4 ■■■2j9-2" 2p J [l + a(^+ l)(2j;+ 1)] 
.1-2=1 
(6) To find ^al we have to evaluate the determinant of (p + l)st order, 
.(46- 
2q-l 
1 
2^1 
+ a 
2q+l 
1 
2^TS 
+ a 
2q + 2p-5 
1 
2q + 2p-l 
+ a 
+ a 
1 
2q+2p-2> 
+ a 
2q + 2p-l 
2q + 4^9-5 
