394 
[850 
850. 
ON LINEAE DIFFEKENTIAL EQUATIONS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xxi. (1886), 
pp. 321—331.] 
1. The researches of Fuchs, Thome, Frobenius, Tannery, Floquet, and others, relate 
to linear differential equations of the form 
d m y d m ~ 1 y . 
p - +p ' dii” + • • ■• + P-y = °. 
or say 
d 
(Po, Pi, •••> l) m y = 0, 
where p 0 , p 1} p m are rational and integral functions of the independent variable x; 
any common factor of all the functions could of course be thrown out, and it is 
therefore assumed that the functions have no common factor. It is to be throughout 
understood that x and y denote complex magnitudes, which may be regarded as points 
in the infinite plane; viz. x, = £ + iy, is the point the coordinates whereof are y: 
and similarly for y. 
2. Suppose x — a is not a factor of p 0 , the point x — a is in this case said to 
be an ordinary point in regard to the differential equation; and let y 0 , y 1} y 2 , ..., y m _i 
du ^ 'll 
be arbitrary constants denoting the values of y, ~^ for the point x = a. 
We can from the differential equation, and the equations derived therefrom by successive 
differentiations in regard to x, obtain the values for x = a of the subsequent differential 
d m y d m+1 y 
coefficients , y x m+ x > •••» sa y these are y m , y m+ i, ... ; viz. the value of each of 
these quantities will be determined as a linear function of y 0) y x , ..., y m - x ; and we 
thus have a development of y in positive integer powers of x — a, viz. this is 
y = Vo + 2/i {x-a) + j-2^/2 {oc-af + ...,