116 Proceedings of the Royal Irish Acadenvj. 



§ 1. The equation ^(/))x = 0. 

 § 1"1. A solution containing tliepj-oper nuniber of constants. 

 Ill the equation D denotes d/cU, and ^ is a polynomial. Obviously the 

 equation is satisfied by 



2.z>= I'^Wx, (1) 



c\ <p (A) 



wherein -i is any analytic function without poles, and the integral is taken 



along any contoui- enclosing all the zeros of ^(X). For (1) gives 



2Tn<p(I>):c= cJ4'W€'''dX, 

 which is zero. 



In particular, -^ may be a polynomial of order lower by unity than that 

 of (p, and with arbitrary coefficients : if ^ is of degree r, the number of these 

 coefficients is r, and this is known to be the proper number of arbitrary 

 constants in the most general solution. 



If the contour in (1) encloses only some zeros of <p, (1) is still a solution, 

 though not the most general. 



§ 1'2. TJie solution for assigned initial values of x and its deriijotives: 

 A definite and natural problem is that of finding a solution so that the 



initial values of x and its derivatives up to the ir - 1)"", inclusive, may 



have given arbitrarily assigned values. 



If the values in question are denoted by -x^,, a-i, . . . Xr.i ; and if 



9 (X) - aX + ar.^y-' + ..., (2) 



then the solution is given by (1), where 



xP (A) = ar (A'-ia-o + X'-'x, ^ . . . + Xr.,) 



+ CEr-l (X''"-.ro + Y'^Xi + . . . + Xr-i) 



+ aiTo ; (3) 



the right-hand member of this may be written 



-4,(I)j-f(X)^ 



B-X ~ t = o W 



To prove that this is the solution it is necessary and sufficient to show 

 further that this expression for x gives to it and its derivatives the assigned 

 values initially. For this purpose we suppose the contour to be everywhere 

 at a great distance from the origin. 



As regards x itself, the initial value of the integrand in (1), with \p as 

 given by (3), tends asymptotically, as A increases indefinitely, to equality 

 with X'^dX . Xi,, the error being of order X"^ ; so that the integral is 2n-ico. 



