Dear Sir
I have to thank you
very much for the last
papers: the proof for the finite
number of the invariants
of a binary quantic is very
much simplified (& indeed
becomes beautifully simple
& easy) if instead of invariants
you consider Seminvariants"
in face writing .... the general
form of an irrational Semiinvariant

as .... and we obtain all the rational
Semiinvariants by considering the like symmetrical forms, or
simply prefixing a Σ to the foregoing expression. But by
16)-2
the equation of differences ... where
the confficients ... 穴あき不明も、おそらくは are functions of Seminouriants: hence
in the expression for S it is only necessary to attend
to forms where in each of the exponents p.g... is at most
..., and the number of such forms is finite. Thus
every Seminvariant is of the form ... where denote
nimber of forms A.B. Is finile, and where s, I
rational r integral funchons of the coefficiants ... of the equation
of differences, or what is the same thing of a finite number
of Seminvariants. I propose to write this one for
the Annalen. I congratulate you very heartly on
this your solution of what has been so long a difficulty.
Believe me dear Sir yours very sincerely
A. Cayley
Cambridge 22nd Janu. 1809.


Dear Sir
I am much obliged for
your two letters: my difficulty
was an a priori one, I thought
that the like process should
be applicable to Semiinvariants
which it seems it is not. &
                for invariants
I now quite see that your
                    to
step from .... to
... is quite right. I
think you have found the
true solution of a great problem.


But the proof of theorem I seems to me to heed revision. I
understand form" to mean homogeneous form: thus ... the
forms .. of theorem I are mere powers of x: and the like for ... &
Suppose to fuse the ideas ... and ...
I is oboiously true, we have ... that is ...; and
hence 4xxx- x which is not homogeneous, &
hence in assuming the theorem to be true for the functions ...
you are in effect assuming it, not for the value n=1, but
for the value n=2: so that you do not in this way pass
from Theorem I, n=1 to Theorem II,n=1.
I think I see my way to a proof of theorem I, starting
with the general case of n variables.
 I wrote one my proof for the Seminvariants (founded
on yours in the Math. Annalen) & sent it to Prof. Klein: it
seemed to me all right, and I think[?] it will appear so to
you when it is published. Believe me dear Sir, Yours
A. Cayley
very sincerely
Cambridge 30th Janu. 1889



ドイツ語のものは、Hilbert の第10問題を書いたもので、次の様に翻刻される。

Beweisen, dass man durch eine endliche Anzahl von Operationen entscheiden kann, ob und wie viele ganzzahlige Lösungen Φ(x,y...)=0 mit ganzzahlige Coefficienten ‹Koeffizienten› besitzt.