Outline of a Theory of Functions of an Abstract Variable 83 
y= Dd apn | > bony} 
1 1 
and on comparing coefficients, after multiplying out, it is seen 
that the coefficients of (6) must satisfy the recurrence formulas 
b(2j@p2} = 1¢2;* 
3 
(8) bj3)@,2) + @[3)bj2) = 0. 
Each equation of this set is of the form bnj@[2] =Ctn], Where a2; 
and ¢(nj are known. Let us write dj; = 1,3) — 21; then from the 
method of successive substitution it is seen that the solution is, 
formally, 
20 
mm 
Bint = Cn 2, Ap] 
0 
which otherwise may be verified directly. However, in accord- 
ance with an explanation already given, the symbolic product 
which constitutes the right hand member of the foregoing equa- 
tion has a meaning only if the factors are multiplicable. By 
reference to the properties of multiplicable vectors that are 
given on p. 9a, it can be readily seen that each coefficient 5;,) 
is well-defined if the aggregate of coefficients aj), a2), - °° are 
all multiplicable among themselves and if the infinite series con- 
verges. The series certainly converges uniformly for || d2)|| 
= Il ti —ay]| <1, but if this inequality is not satisfied, what 
then may be said concerning a solution? 
Let us consider the function Dxp;", from which, by con- 
tinuation, we obtain an extended regular function. This latter, 
which we denote by (1 —X19;)~!, is well-defined and regular 
everywhere within its domain of existence except at its singu- 
lar points. Because of this, and the fact that functional rela- 
tions persist in continuation, we may write 
Bin) = Cnj(Gq21)* 
in place of the last equation, and assert that this is certainly a 
solution if aj.) is not a singular point of (x (2})~*. On the other 
hand if @{2; is a singular point of the function, a solution may or 
May not exist, as we know from the theory of linear functional 
equations. 
* Here 11, is that element of Ep) which satisfies the equation Ipiy=y 
for all y. It is easily seen with the aid of-P.22 that 1,2) is symmetric. 
It is also clear 1pjapj =aps}1 pj =a pp) whatever af} may be. 
ti 
