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nupber of terms, each sum in the parentheses is a pelynomial-
Multiplying these polynomials, any term of the result is formed by
taking as a factor one term from each of the parentheses. Every

term of the result then is a product of factors of the type
1 Tl bl Ne. i L o !
T {fi[-& . ]} with a condition on the indices, and finally

pultiplied by (-1)™ m! . The condition on the indices is that
S(mgrgy + ***c 4t )=h, With (14) we had by definition

m=j+k+t " +n-1, Sincé we must have j, + °*°° 4 Ji, = J we see
that the sum of a cert;;in first group of n's must equal j;
similarly the sum of a second group must equal k3 etc. Therefore
Ry +Mp 4 ****4ny;~1=m, In (14) we take the sum of all terms
which satisfy the condition j+2k+ ***+4 4r=h, and therefore now
take the sur of all terms satisfying the equivalent condition
Smelrg + *o*v 4 t) = h,

Turoren VII, If the elevents of the array (1) satisfy the
hypothesis of Theorem 1V amd |II = 1| < 1-¢ when |x| < Ry, then
log IT = Zu, where
(15) w, = 2 % {p[ % ...%\3]}"‘...... {p[%‘ ...%\4 ]}""
nhere the summation is taken for all values of the indices for

which 3ng(r¢+ *°**+ty)=h, and where by definition my 4 ¢ +ng=-1= m

Using formula (7) we obtain w as a power series which con
verges for |x| < R,. Using (15) we obtain v as a power series which
converges for |x] <R, £ R,. Then for |x| < R, the series are
ideptically equal. When the s's and ¢'s are expressed in terms of
the elements cf (1), this equality is an identity in the elements,
New if we replace the elements of (1) by those of (1') as described
in the proof of Theorer V on page 20, and collect the coefficients
of each power of the F's the equation is an identity in the Vs,

and each side of the identity is a pclynomial in the Y's. By