130 
PROF. H. F. BAKER ON CERTAIN LINEAR 
Preface. 
Part II. of the present paper was written, very much in the form in which it is 
now presented, in the summer of 1913, and began with the remark in §11, which 
appears to disprove a statement made by Poincare in regard to the convergence of 
an astronomical series. It was laid aside partly because a good deal of the work is 
only of the nature of elementary algebra, partly because the matrix notation 
employed does not seem to find favour in its application to differential equations. 
Various circumstances have, however, led me to take up the matter again, and my 
original conviction that the method of Part II. is of importance has been strengthened 
by comparing it with the less formal methods which, for the sake of introducing the 
subject, I have followed in Part I. I hope, therefore, that the following exposition 
may be thought worth while. Part III. has only the value of a concluding remark. 
The table of contents above may serve to give an idea of the scope and arrangement 
of the paper. 
PART I. 
§ 1. Consider a linear differential equation 
U^+Y^+WX = 0, 
CLt QjT 
where U, V, W are power series in a small quantity, A, of the forms 
U = u -F Aiq + \’'u 2 T" • • • , 
V = v + Xv v + A \’ 2 + ..., 
AV = \w 1 + \ 2 iv 2 +..., 
in which each of u r , v r , w r is a linear function of 
r, f 4 -;, 
l denoting e\ Thus each of u 2n , v 2n , iv 2n will contain a term independent of we 
speak of these as the absolute terms. It is important that AV contains no term 
in A" ; and it is assumed that the quantity v/u, which is independent of £, is not a 
positive or negative integer, and that u, v are not both zero. 
AVe prove that if the absolute terms in AV, that is the absolute terms in 
w 2 , w i} w 6 , 
be suitably determined, the differential equation possesses a solution of the form 
X — l + x<pi + A“0 2 t • • •, 
