MATHEMATICS: F. R. MOULTON 353 
It follows from (1) that the explicit expressions for bi, gi''^\ gi^^\ . . . 
are 
ot oxi d«2 
(17) 
n! Lot oxi 0x2 
where the power indicated in the last equation is symbolic such that 
(18) 
X2 / 
(flQ + Wl + 
• = n). 
are 
After the partial derivatives have been formed, t, Xi, X2, 
replaced by to, Xi\ X2^\ ■ • • respectively. 
Since the transformation (15) is linear, g^"^ depends only on fi"\fi"'^^\ 
. • • . It follows from the fact that f\''\fl"'^^\ • • • are domi- 
nated by AriS'\ AfiS"'^^, - • • respectively that 
S" 
df" dxi"' bx2' 
/ = |/ol 
1 ~S 
It is easily found that 
n \ Co " cr C2 
Now let 
The series 
(1) _ ^0 
(1) 
C2 
1 — So 1 — So 1 — "5*0 
converges if 
|r|gr„(l-5„), |yi|gn(l-5o), \yi\^r,{l-So) 
and Irk 1 if 
roil-So) 
(19) 
(20) 
