768 REPORT—1863. 
stock on railways, and it is stated that springs applied in this manner effect 
an increased durability in Staffordshire tyres of 50 per cent. over Krupp’s 
cast-steel tyres without the springs. 
The estimated annual value of the steel manufactures of the district is 
about £100,000, giving employment, at the present time, to about 300 
persons, and consuming annually about 15,000 tons of coals. There are in 
the district nine converting furnaces and fifty-two cast-steel melting-furnaces, 
The following is a list of the firms possessing those furnaces :—Messrs. John 
Spencer and Sons, Newburn, six converting and thirty-six melting-furnaces ; 
Messrs. Cookson and Co., Derwentcote, one converting and six melting-fur- 
naces; Messrs. Faweus and Co., Swalwell, two converting and six melting- 
furnaces ; Messrs. Fulthorpe and Co., Dunston, six melting-furnaces, 
As far as can be ascertained, it would appear that the number of persons 
employed in this trade in 1838 would be from seventy to eighty, and the 
weight of steel produced annually at that time would be about one-ninth the 
quantity now produced. ‘The prices of steel range from about £18 to £112 
per ton, according to the description, the quantity, and the size. 
This district is highly favourable for the development of the manufacture 
of steel of the best quality, owing to the facility and cheapness with which 
a supply of iron can be obtained from Sweden—treights frequently being as 
low as 3s. 6d. per ton—and also owing to an abundant supply of cheap fuel 
and labour in the neighbourhood. ‘The business requires, however, the most 
vigilant attention of thoroughly practical and experienced persons in its 
management to attain any considerable amount of success. 
Report on the Theory of Numbers.—Part V. By H. J. Srernen 
Surru, M.A., F.RS., Savilian Professor of Geometry in the Univer- 
sity of Oxford. 
120. Geometrical Representation of Forms of a Negative Determinant.— 
Before quitting the subject of binary quadratic forms, we have still to men- 
tion several investigations of great interest, relating chiefly to forms of a 
negative determinant. We shall first refer to the geometrical considerations 
which Gauss has employed to illustrate the nature of these forms *. 
Let an infinite plane area be divided by two systems of parallel lines into 
similar and equal parallelograms. The vertices of these parallelograms we 
shall call nodes; and we observe that every system of nodes possesses the 
characteristic property, that if it be displaced without rotation in its own 
plane, so as to bring any one node into a position originally occupied by any 
other node, then every node will also occupy a position originally occupied by 
another node; and the system in its second position will entirely coincide 
with the system in its original position. From this property we infer that 
the system of nodes admits of an infinite number of parallelisms besides the 
given parallelism; 7. e. that it may be regarded, in an infinite number of dif- 
ferent ways, as dividing the plane area into similar and equal parallelo- 
grams, For let O and O' be any two nodes such that no node lies on OO' 
between O and O'; let P be one of those nodes which lie the nearest to the line 
* See Gauss’s review of Seeber’s ‘* Untersuchungen iiber die Higenschaften der ter- 
niiren quadratischen Formen,” in the Gottingen ‘ Gelehrte Anzeigen’ for 1831, No, 108, 
or in Crelle’s Journal, vol. xx. p. 312; also Lejeune Dirichlet, Crelle, vol. xl. p. 209. 
