■536 EIGHTH PACIFIC SCIENCE CONGRESS 



The method of solving equations (63) and (64) and the evaluation 

 of the integral: 



00 



will be discussed in the following section. 



9. Integration of the Differential Equations 

 The next step will be to solve the ten differential equations (63) 

 and (64). Let any one of these equations be 



^'^ - 108232377^-^^ - 14997110— -f brj'Y i- c = (70) 



dX^ dx^ dX 



where h and c are constants assigned to each of these ten equations. 

 Since a particular solution of this equation is 



^'-~ ~~' (71) 



the general integral of (70) will be of the form: 



clb 

 Y{X) - Ae'^'^ + 5e^^ + Ce^'^ + De^^ — — ^— (72) 



where «, p, y and S are the four roots of the algebraic equation: 



(T* — 10823237?'a- — 14997110a + 67^2 = q (73) 



and A, B, C, D are constants to be determined according to the con- 

 ditions: 



y(0) = 7(1) = 0; 



r(0) = y'(i) = 0. (74) 



The equation (73) has four roots a, (3, y and S for any given value 

 of 7], and A, B, C and D all depend upon a, /?, y and S- 



The parameter rj varies from Oi to co, and the values of a, ^, y, S 

 all depend upon -q. When -q is very large, these four roots are approxi- 

 mately 



a = + 1040.3075,;, 



p = — 1040.30757?, 



y = + (6/1082323)'/^, 



8 = — (&/ 1082323) Vs. (75) 



As 7] decreases, «, p, y and S also change gradually. For -q less than 



a certain value between -q = 0.4 and 7? = 0.3, ^ and S become complex 



conjugate. As -q approaches 0, a, /? and S approach finite values, while 



y decreases indefinitely as 0077". Thus we have, when 77 — > 0. 



a = 2p, 



P = — p + iq, 



8 = — p — iq. (76) 



