196 
PACIFIC SCIENCE, Vol. IX, April, 1955 
Practically, the equation corresponding to (53) is 
w 
+ 8 
-6 
+4 
-2 
-3 
20 -r 
+ 18 
-12 
+ 6 
+ 5 
-15 
69-r 
+ 20 
-10 
-7 
+ 21 
-35 
i48 -r 
+ 14 
+ 9 
-27 
+45 
-63 
257- : 
210830400 - 84156468^ + 4625225f 2 - 7877ir 3 + 495f 4 - f 5 = 0 (57) 
where 
r = £/ (4.38649084 X A) (58) 
and the five roots of (57) are 
fi = 2.9641943, 
r 3 = 26.7072817, 
r 5 = 74.3718483, (59) 
r 7 = 146.456904, 
r 9 = 244.499771. 
For even series we have, corresponding to (57), (58), and (59), 
7— y 
+ 16 
-12 
+ 8 
-4 
-8 
43 — 7 
+ 24 
-16 
+ 8 
+ 12 
-24 
111 — 7 
+ 24 
-12 
= 0 
-16 
+ 32 
-48 
211 — 7 
+ 16 
+ 20 
-40 
+ 60 
-80 
343 - 7 
5217079023 - 593115237 7 + 16671798+ - 172458+ + 715 7 4 - 7 5 = 0 (60) 
where 
y = £/ (4.38649084 X %) (61) 
and 
7 2 = 12.8567852, 
7 4 = 51.494636, 
76 = 116.1536052, (62) 
7s = 207.4231320, 
7 io = 327.071842. 
Differential Equations for Y's and Their Solutions 
From the numerical computations described above, the numerical coefficients of the 
differential equations for F m (A; rj) were determined as follows: 
D(Y i) + 11514116+Fi - 0.4037686 X 10 20 = 0, 
D(Ys) + 103741 765?7 2 F 3 - 6.3254121 X 10 20 = 0, 
D(Y b ) + 28889000 8+F 5 - 204.1449265 X 10 20 = 0, 
D(Y 7 ) + 568897198^ F 7 + 311.4127463 X 10 20 = 0, 
D(Y 9 ) + 949734910+ F 9 + 257.7413034 X 10 20 = 0 
( 63 ) 
