TRANSACTIONS OF SECTION A. 307 
a 
Introduce symbolic operators 4 y= ral (k—h)dva+ as h) 
x 
y 
1 
A,= ‘| (—k)dyB + 7) 
y 
@ 
. 
ll 
~ 
+ 
Pa 
Db 
< 
+ 
fs, 
co] 
b 
oN) 
—" 
eo 
+ 
Then a symbolic solution appears as 
x y 
= [ax = [aay 
ec O,Hz)+e 0, H(u) 
where 4, w are arbitrary functions of their arguments. Here the arbitrary elements 
enter in an infinite series of their derivative. 
We may deduce a form of solution in which ¢, p enter in finite terms, namely— 
ay 
= Bdx (6 —A) 
Ty sey Sle 
_— Bu. a 
+e woe,]° 
y=0 
(8. “Meda ] 
eal dt 
Applying these results to the equation S=z we obtain a solution 
y x 
2= | p(t). J {iy — t)a} dt + | V(t).Jd {(x—t)y} dt+C.J(ay) 
) to) 
u wu wr 
where C is a constant and J(u) =1 + ap + 1 #oaes (n!)? + 
2. Properties of Algebraic Numbers Analogous to Certain Properties of 
; Algebraic Functions. By Professor J. C. Freups, F.R.S. 
Suppose ¢ to be a root of an integral algebraic equation f(x) = 0 of degree n in x 
and irreducible in the domain of the rational numbers. Where p is a prime f(x) 
may, however, happen to be reducible in the domain of the p-adic numbers. As- 
suming the number of the irreducible p-adic factors to be r we write f(x) = f,(x) 
....f,(z), where the coefficients of the powers of x in f,(x),... f(x) are p-adic numbers. 
Consider R(e) any rational function of «. It may be written as a polynomial 
of degree n—1 in ¢ and satisfies an algebraic equation F (X) = 0, where we have 
F(X) = F(X)... F(X). The factors F(X), ... F(X) here have p-adic 
coefficients and are co-ordinated with the factors f,(z), . . . j.(v) of f(z). The 
degrees of the factors f,(z), . . . f,(x), as also those of the factors F(X), .. ., 
F,(X) will be certain integers n, .. ., n, respectively. The constant terms in the 
factors F(X), . . . F(X) we name the p-adic partial norms of R(«). The 
order numbers relative to p of the p-adic partial norms of R(<) divided by m,.. ., 
x 2 
