396 ON LINEAR DIFFERENTIAL EQUATIONS. [850 
this is so, we substitute in the differential equation for y the value in question 
x p E(x), thus obtaining a series 
n 0 xp- 6 + + ..., 
(where 6 is a determinate positive integer depending on the negative powers of x in 
the equation); the coefficients fi 0 , i! 1} ... are functions of p of an order not exceeding 
m, and contain also the coefficients E 0 , E u E 2 , ... linearly; in particular, il 0 contains 
E 0 as a factor, say its value is = A , 0 II 0 . The series should vanish identically. Supposing 
that II 0 contains p, then we have II 0 = 0, an equation of an order not exceeding m 
for the determination of p. For any root p — po of this equation, E 0 remains arbitrary 
and may be taken = 1 ; the equations ilj = 0, il 2 = 0,... then serve to determine the 
ratios to E 0 of the remaining coefficients E 1} E 2 , ...; and we thus have the solution 
y — x p <> (1 + E x x + E 2 x 2 + ...), where p a and the coefficients have determinate values. 
6. I stop to notice a curious form of illusory solution ; the assumed form of 
solution is 
y — x p (. •. P E_%x " P E_j x 1 P E 0 p E-yX P ...), 
the series being a double series extending both ways to infinity, or say a back-and- 
forward series ; we have here a series of equations 
...n_ 2 =0, n_ x = o, n 0 = o, n 1 = o,..., 
which leave p undetermined, but determine the ratios of the several coefficients to one 
of these coefficients, say E 0 \ or taking this =1, we have a solution 
y = x p (... P E_ 2 x~' 2 + E^x* 1 + 1 + E x x + E 2 x 2 + ...) 
where the coefficients are determinate functions of the arbitrary symbol p. Such a 
series is in general divergent for all values of the variable, and thus is altogether 
without meaning. As a simple instance, take the differential equation — y = 0, 
which is satisfied by 
y—\...(p — 1 ) px p ~ 2 + px p ~ 1 + x p + 
x p+1 
P + 1 
+ ... 
see my paper, Cayley, Note on Riemann’s paper, “ Versuch einer allgemeinen Auffassung 
der Integration und Differentiation,” Werke, pp. 331—344; Math. Ann. t. xvi. (1880), 
pp. 81, 82), [751]. 
7. A more general form of integral is Thome’s “ normal elementary integral,” 
y=e w x p E(x), where w is = a finite series “¡Pinegative powers of a: (a a 
positive integer, = 2 at least). To discover whether such a form exists, observe that, 
writing for a moment ^ = y', and so for the other symbols, we have -=«/+-+ , 
° dx * j > y x E(x)