526 : REPORT—1862., 
arrangement of the genera and classes is in accordance with the construction 
of Gauss, explained in the preceding articles; and the position of each class 
in the arrangement is indicated by placing opposite to it, in a separate column, 
the term to which it corresponds in the symbolic formula (such as |K| or 3 x |K}) 
which forms the type of the arrangement, To the two Tables of positive and 
negative determinants Mr. Cayley has added a third, containing the thirteen 
irregular negative determinants of the first thousand. 
In a letter addressed to Schumacher, and dated May 17, 1841, Gauss 
expresses a decided opinion of the uselessness of an extended tabulation of 
quadratic forms. “If, without having seen M. Clausen’s Table, I have 
formed a right conjecture as to its object, I shall not be able to express an 
opinion in fayour of its being printed. If it is a canon of the classification 
of binary forms for some thousand determinants, that is to say, if it is a 
Table of the reduced forms contained in every class, I should not attach any 
importance to its publication. You will see, on reference to the Disq. Arith. 
p- 521 (note), that in the year 1800 I had made this computation for more 
than four thousand determinants ” [viz. for the first three and tenth thou- 
sands, for many hundreds here and there, and for many single determinants 
besides, chosen for special reasons]; ‘‘ I have since extended it to many others ; 
but I have never thought it was of any use to preserve these developments, 
and I have only kept the final result for each determinant. For example, for the 
determinant —11,921, 1 have not preserved the whole system, which would 
certainly fill several pages *, but only the statement that there are 8 genera, 
each containing 21 classes. Thus, all that I have kept is the simple state- 
ment viii. 21, which in my own papers is expressed even more briefly. I 
think it quite superfluous to preserve the system itself, and much more so to 
print it, because (1) any one, after a little practice, can easily, without much 
expenditure of time, compute for himself a Table of any particular determi- 
nant, if he should happen to want it, especially when he has a means of 
yerification in such a statement as vill. 21; (2) because the work has a cer- 
tain charm of its own, so that it is a real pleasure to spend a quarter of an 
hour in doing it for one’s self; and the more so, because (3) it is very seldom 
that there is any occasion to do it....... My own abbreviated Table of the 
number of genera and classes I have never published, principally because it 
does not proceed uninterruptedly.” + Probably the third of Gauss’s three 
reasons will commend itself most to mathematicians who do not possess his 
extraordinary powers of computation. An abbreviated Table of the kind he 
describes, extending from —10,000 to +10,000, would occupy only a very 
limited space, and might be computed from Dirichlet’s formule for the 
number of classes (see Art. 104), without constructing systems of repre- 
sentative forms. But it would, perhaps, be desirable (nor would it increase 
the bulk of the Table to any enormous extent) to give for each determinant 
not only the number of genera, and of classes in each genus, but also the 
elements necessary for the construction, by composition only, of a complete 
system of all the classes. For this purpose it would not be necessary to 
specify (by means of representative forms) more than 5 or 6 classes,’ in the 
case of any determinant within the limits mentioned. 
* Mr. Cayley’s Table of the first hundred negative determinants occupies about four 
pages of Crelle’s Journal; the determimant —11,921 would occupy about one page, 
+ Briefwechsel zwischen C. F. Gauss und H. C. Schumacher, yol. iv. p. 30. 
