198 
PACIFIC SCIENCE, Vol. IX, April, 1955 
Substituting the functions Aff(x; 77), N 2 (x; 77), , Ni 0 (x; 77) thus obtained in (30), we 
shall obtain the expression for \J/ i(x, y; 77). Further substitution in (29) will give the solution 
of the problem as 
iK*, y , *) 
2D^ M ^y) ■ J 
N m (x; 77) cos 
dr]. 
(69) 
The method of solving equations (63) and (64) and the evaluation of the integral: 
/ N m (x; 77) cos (jjy v) drj 
2 
will be discussed in the following section. 
Integration of the Differential Equations 
The next step will be to solve the 10 differential equations (63) and (64). Let any one 
of these equations be 
?4 J7- 7 2 T7- 1 XT 
+ - 1082323tj 2 - 14997110 + + b v 2 Y + c = 0 (70) 
aX 4 aX 1 aX 
where h and c are constants assigned to each of these 10 equations. Since a particular 
solution of this equation is 
(71) 
the general integral of (70) will be of the form: 
F(X) = Ac aX + Be pX + Cc yX + Dc SX - + (72) 
77 
where a, (3, 7, and 8 are the four roots of the algebraic equation: 
<7 4 - 108232377V 2 - 149971100- + V = 0 (73) 
and A, B, C, D are constants to be determined according to the conditions: 
F(0) = 7(1) = 0; Yi(0) = 7((1) = 0. (74) 
The equation (73) has four roots a, (3, 7, and 8 for any given value of 77, and A, B, C, 
and D all depend upon a, (3, 7, and 8. 
The parameter 77 varies from 0 to «>, and the values of a, (3, 7, 8 all depend upon 77. 
When 77 is very large, these four roots are approximately 
a = +1040.307577, 
(3 = -1040.307577, (75) 
7 = +(6/1082323) 4 , 
5 = - (6/1082323) K 
As 77 decreases, a, (3, 7, and 8 also change gradually. For 77 less than a certain value be- 
tween 77 = 0.4 and 77 = 0.3, (3 and 8 become complex conjugate. As 77 approaches 0, a, 
(3, and 8 approach finite values, while 7 decreases indefinitely as « rf. Thus we have, 
when 77 — > 0, 
a = 2p, 
(3 = ~p + iq , 
7 = br] 2 /Sp'\ 
8 = —p — iq 
(76) 
