851] 
ON LINEAR DIFFERENTIAL EQUATIONS. 
407 
Set (4) gives 
p 3 = — a +p 4 , q 3 = a + a 2 r 3 = — (a" + 3aa + a 3 ) 
-p 4 a —p' + p 4 (a 2 + a) + 2p/a + p" 
+ p 4 , -q 4 a -qi 
+ r 4> 
0 = a" + 4aa" + 6a 2 a' + 3a 2 + a 4 
— p 4 (a" + Seta' + a 3 ) — pi (3a' + 3a 2 ) - 3p 4 "a — pi" 
+ qi (a + a 2 ) + 2,q 4 ’cc + qi' 
— r 4 a — ri 
+ s 4 , 
which differ in form from the equations belonging to the reverse order in containing 
the derived functions pi, pi', pi”, qi, ... of the coefficients. 
9. Taking p 4 , q 4 , r 4 , s 4 as known, we have a determined by a differential equation 
(not linear) of the third order; and a being known, we know p 3 , q 3 , r 3 . We then 
have /3 determined by a differential equation (not linear) of the second order; and 
/3 being known, we know p 2 , q 2 . We then have 7 determined by a differential 
equation (not linear) of the first order; and 7 being known, we know p 1} that is, 8.