Colloquium publications.

122

THE CAMBRIDGE COLLOQUIUM.

37. By the definition of the Akj's there exists a set of homologies
(13) rki 'V - ijAkj (i = 1, 2, *.., ).
j=1
If we substitute (12) in (13) we obtain
(14) Ik 1japrk (i - 1, 2,..., t).
j=1 p=l
For i = 1 this gives the homology
(141) ( Z fljol - l)rk' + ( E ljoj2)rk2 +. '
j=1 j=1
+ ( E EjaJ.)Fr k 0.
j=1
If this homology is multiplied by the last coefficient of torsion
all the last rk terms drop out in virtue of the homologies (4) of
~ 30, since all the k-dimensional coefficients of torsion are factors
of t1k. The remaining terms would give a homology connecting
rk', I k2, *, rk k-l, contrary to ~~ 30 and 34, unless the first
Pk- 1 coefficients in (141) were zero. Hence these coefficients
are zero. If (141) is multiplied by the coefficient of torsion
tlk all terms of (141) drop out except the last. Hence the
last coefficient of (141) is zero. If (141) is multiplied by t,_2k,
all terms of (141) except the next to the last drop out. Hence
the next to the last coefficient of (141) is zero. And by a similar
argument all the rest of the coefficients of (14) are zero.
The same reasoning can be applied for all values of i (i = 1, 2,
~*,,u) to show that the coefficient of every rPk in (14) is zero.
Hence
(15) ij 1. 1 i- =I,
where I is the identity matrix of y rows and columns. Hence the
determinant of IcjpI is unity and [|ijll is the inverse of
I 11icil
From this it follows in an obvious way that all homologies
among the Ak 's are linearly dependent on the tk homologies,
(16) tik( Z Pk-l+ijAkj) - 0 (i = 1, 2,.., Tk),
j=1