116 PROCEEDINGS OF THE AMERICAN ACADEMY. 



this case v and/ are both zero. Let us therefore assume a solution for 

 a: as a series 



x = a(l 4- clt 3 + 0r 4 4- 7 t 5 + . . .) 



in r. The equations for the determination of the coefficients are 



36a 2 /c 4- 3a = 0, 144a/3/c + 18a 2 + 4/3 = 0, 



(144/3 2 + 240a7> + 60aj3 4- 5? = 0, . ., . 



Hence 



or 



a = 0, = 0, 7 = 0,... 



11 11 



a =~TFT> = 7Tr5>" 7= — 



12k' ^ 64k 2 ' ' 3840k 3 '' 



Thus one solution for x is .r = a, the particle remains at rest in the 

 initial position, and the other solution is 



/ t 3 r 4 llr 5 \ 



X=a V 1 ~l^ + 64K 2 -3840^ + ----> (9) 



the particle moves with an acceleration toward the origin, as was 

 foreseen under G above. 



If this solution in series is valid, that is, if the series converges, it is 

 at any rate of no value for purposes of calculation unless r is exceed- 

 ingly small, that is, for intervals of time which are very small com- 

 pared with the period of monochromatic light (r = 2ir). The value of 

 the series is first to show how the motion starts and second to show 

 that there is a possibility of motion, that is, that there is a solution of 

 the equation other than the solution x — a with the given initial 

 conditions. In discussing this question we shall follow the method 

 given by Picard. 8 



Let the equation (6') of the second order be written as two simul- 

 taneous equations of the first order, namely, 



..^Y+.fe+.W £ — 0. do) 



K dr ) \dr ' J dr 



These may be expressed in solved form as 

 dx dv v 1 



y- = v, -j- ' ^v 2 — 4kvx. 



at (It 2k 2k 



The fundamental theorem on the existence of integrals fails for those 



8 E. Heard, Traite d'Analyae, 3, chap. 3. 



