DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 395 



C... = coefficient of p~ in (p? + i, pV + $, />V U + . . . y 



{p + ( W ' + i P Y + i, pV" + ...))' 



and therefore 



C 



while for values of m greater than s 



When all these values are substituted in (iii.), it becomes 



(n-l)e *- P, 



and, therefore, dividing each side by the coefficient of Q, and retaining only first 

 powers of e, we have 



= P,(i- V) 



after a slight reduction. This equation is true for the values s = 2, 3, . . . , n ; and 

 particular cases, to be used immediately, are 



Q s = P 8 (1 - 3c M ') - 3^ U P S - i (n + 1) e^, 



Q 4 = P 4 (1 - 4 C/ x') - 6 eM P 3 - (n + 5) ^P, - ^ (n + 1) /*', 



Q 6 = P 6 (1 - 5 /A ) - 1 Oe/xP 4 - |(n + 7) V P 8 - | (n + 3) e M "P 2 - | (n 



Q, = P 6 (I - 6 CA t') - 15 e/A P 8 - | (n + 9) V P 4 - 5 (n + 



Q 7 = P 7 (1 - 7,/x 1 ) - 21 VP, - S (+ 1 1) V P, - V (n f 



- V (n + 3) M 'P 3 - 7 (n + 2) e ^P 2 - f (n + 1 ) 

 3 2 



