Validity of Approximate Solution of a Velocity Equation. 461 



for values of r above unity and the typical number (4) becomes — 



R;=EM[E E r +E I E J -,..+R J - I B ] (7) 



Hence, using the initial conditions — 



E r =f ()£ ^[E E r + E I E ) - I ...+Er- I Bx + E r Eo]^ (8) 



It is here to be noted that E o = ; and hence when the expansion (2) 

 is applicable the values of the coefficients can be determined by 

 quadratures from (8). 

 The solution is — 



\£ rt i*[l + a 2 \E 2 + a*\ 2 R 3 +...] (9) 



To discuss the convergence of (9). 



From the values of E and E„ and the mode of formation of the 

 subsequent coefficients, it is clear that the E's are all positive for 

 positive values of t. 



£ a z t j g one f ac tor in the solution, and will occur to higher powers 

 in the other factor ; so that, if we include in our consideration large 

 values of t, a necessary condition for the convergence of the series is 

 a x negative. 



Put a I =—a 1 where a z is positive. 



Applying this result we have — 



E o = 

 R x = l 



f* 1 



E 2 = e-i'.R^E^- 



J o «i 



R. 



■* 1 



MEA + R 2 Ex)< (K 2 K X + KxK 2 ) 



«i 



E w <^ | K^K; + K„_ 2 K 2 ... + K 2 . K n _ 2 + KJv^ J 



where the K's satisfy the relations — 



K o = 

 K 1 = l 



K 3 =^(K 2 K I + K I K 2 ) 



K, *= 1 (K^K, + K,._ 2 K 2 . . . + K 2 K r _ 3 + K.K^) 



(10) 



(11) 



