Validity of Approximate Solution of a Velocity Equation. 463 



Hence conditions ivhich are certainly sufficient for the applicability 



of this form are — 



(1) a x negative. 



a-L I 

 4aJ. 



(fl)X< 



(For the case considered in §1 this gives X<tt.i whereas we know 



that the range of availability of the method is bounded by A.<t). 



§3. Apply the same method to the more general equation — 



dy 



-f = a x y + a 2 y 2 + a 3 y3 + 



subject to the initial condition y = \ when t = Q. 

 Try a solution of the form — 



(13) 



(14) 



,(15) 



y = T + \Tx + \ 2 T 2 + 



Substitute in (13) and equate coefficients of X — 



.:T =a 1 T + a 2 Tl + a 3 Tl+ ... 

 T^flfeT, + fl^ToT, + T X T ) + a 3 . 3T;T, + . 

 T; = a,T 2 + a 2 (T T 2 + T?) + a 3 . 3T; . T + . . 

 &c. 



The initial conditions for the different T's are — 



T o = 0, Tx = l, T 2 = 0, T 3 = ... when tf = 0. 



Substitute T r = e°«* S — 



.\S o =0 



S I = 1 



SU«x* . a 2 (S S 2 + s x ) + e°^ . a 3 (3S s;) + . . . 



S; = #* . a 2 (2SxS 3 ) + £ 2a -*(3S S 3 + 6SxS 2 S ) + . . 



Once more it is to be noticed that the coefficients T or S can be 

 immediately evaluated by quadratures, since the coefficient of T r in 

 the equation for T' r is zero, and the earlier T's are calculated in 

 succession. 



As in the case discussed in §2, a first condition for convergence 

 will be a x negative, equal to - a z (say) where <*! is positive. Follow 

 the line taken in §2, putting | a 2 \ —a 2 , \a 3 \ =a 3 , &c. ; we then find 

 that each of the S's has, for all values of T, a modulus not greater 



(16) 



