Formula (18a) give the initial conditions required for the solution to be of "principal modes" type.Appendix.-If in the elements of the sequence { an . n = O y1, I 2, ... , where X-n = an, the element a,, and then a-1, is dominant, it is easily seen that the dominant sum of the double series of (3) is given by 3a,2an.2 + 6 lal 12an + 3&12an+2, (Al) where n is any integer except n = 4 1, 4 3. For these exceptions, the dominant terms of the double series are 3a~a12 forn = 1, 3alaI2 forn =-1, a3 forn = 3,& 3 forn = -3. (A2)
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