564 
however, only a few authors have attached any import- 
ance. On investigation Mr. Cheyne found that the 
tubercle-bacilli were, unless when present in large num- 
bers, only found in or among these epithelioid cells, and 
that the tuberculous nodules first begin by the entrance 
of bacilli into these cells and the subsequent development 
of the epithelioid elements. Surrounding these epithelioid 
cells a slight amount of inflammation occurs, giving rise 
to the small-celled growth around the tubercle, which is 
generally regarded as the growing part of the tubercle. 
This Mr. Cheyne denies, asserting that it is merely in- 
flammatory tissu2, and that the essential elements of the 
tubercle are the epithelioid elements in its centre. In 
the lungs these cells seem to be derived from the alveolar 
epithelium, in the liver often apparently from the liver 
cells, but in other organs and also sometimes in these 
from the endothelium of the lymphatics and blood-vessels. 
In phthisis the bacilli were found at the margin of 
cavities and in the epithelioid cells surrounding the cheesy 
matter. Mr. Cheyne concludes that in phthisis the 
bacilli, inhaled into the alveoli, develop in the alveolar 
epithelium, cause accumulation of epithelial cells in the 
alveolus, and inflammatory hypertrophy of its walls. 
Thus the bacilli are practically shut off from the circula- 
tion and acute general tuberculosis cannot occur. The 
two extremes of phthisis are considered—the very rapid 
form or caseous pneumonia, and the slow form or fibroid 
phthisis. In the former the bacilli grow rapidly, are 
fairly numerous, and the lung rapidly breaks down; in 
the latter the bacilli grow slowly and with difficulty, and 
hence extensive fibrous formation occurs. 
There are many other points of interest in this research 
to which we cannot allude, but which will be found at 
length in the Report. The Association is to be congratu- 
lated on having chosen such a fertile subject for their first 
report, and we hope that they will continue to encourage 
similar work. 
PROFESSOR H. J. S. SMITH AND THE REPRE- 
SENTATION OF A NUMBER AS A SUM OF 
SQUARES 
HE award of the great Mathematical Prize of the 
French Academy to the late Prof. H. J..S. Smith 
may have the effect of drawing the attention of 
mathematicians to the wonderful extent and value of his 
researches on the Theory of Numbers. Probably no 
more important or remarkable mathematical investiga- 
tions have ever appeared in this country than his memoirs 
on systems of linear indeterminate equations and con- 
gruences and on the orders and genera of ternary 
quadratic forms and of quadratic forms containing more 
than three indeterminates, which were published in the 
Philosophical Transactions for 1861 and 1867 and the 
Proceedings of the Royal Society for 1864 and 1867. The 
results contained in these papers are by far the greatest | 
additions that have been made to the Theory of Numbers 
since it was placed on its present foundation by Gauss in 
the ‘‘ Disquisitiones Arithmetica.” The subject for which 
the prize was awarded to Prof. Smith was that of the 
theory of the representation of a number as a sum of five 
squares, and of this question as well as that of the cor- 
responding one for seven squares he had given the 
complete solution in the Proceedings of the Royal Society 
for 1867 (vol. xvi. p. 207). The words with which Prof. 
Smith introduced his statement of the solution of these 
important questions are as follows :— 
‘‘The theorems which have been given by Jacobi, 
Eisenstein, and recently in great profusion by M. Liou- 
ville, relating to the representation of numbers by four 
squares and other simple quadratic forms, appear to be 
deducible by a uniform method from the principles indi- 
cated in this paper. So also are the theorems relating to 
the representation of numbers by six and eight squares, 
NATURE 
[April 12, 1883 
which are implicitly contained in the developments given. 
by Jacobi in the ‘Fundamenta Nova.’ As the series of 
theorems relating to the representation of numbers by 
sums of squares ceases, for the reason assigned by 
Eisenstein, when the number of squares surpasses eight, 
it is of some importance to complete it. The only cases 
which have not been fully considered are those of five 
and seven squares. The principal theorems relating to 
the case of five squares have indeed been given by 
Eiseastein (Cred/e’s Journal, vol. xxxv. p. 368) ; but he has. 
considered only those numbers which are not divisible by 
any square. We shall here complete his enunciation of 
those theorems, and shall add the corresponding theorems 
for the case of seven squares.” 
In the announcement of the subject for the prize in the 
Comptes Rendus in February of last year, reference was 
made to the work of Eisenstein, but the fact that his 
solution had fifteen years before been completed by Prof. 
Smith—who had also solved the problem in the case of 
seven squares, the whole being only a corollary from the 
general principles contained in his memoirs—seems to 
have escaped the attention of the propcsers of the 
subject. In the paper in the Proceedings of the Royal 
Society the results only for the case of five squares 
and seven squares are given, the demonstrations being 
omitted ; and accordingly, when the subject for the prize 
was announced, Prof. Smith followed the only course 
open to him, and communicated to the Academy his 
demonstrations for the case of five squares. 
All who knew Prof. Smith will understand how uncon- 
genial to him was the idea of becoming a competitor for 
the prize, but under the circumstances he had no choice. 
It is a singular tribute to Prof. Smith’s mathematical 
powers, as well as a curious episode in the history of 
mathe natics, that the French Academy should have 
chosen as the subject of the “Grand Prix”—thereby 
indicating their opinion of its importance in the advance 
of the science !—a question that had been solved already 
fifteen years before as a corollary from more general 
principles. 
The state of the question of the number of ways in 
which a number can be expressed as a sum of squares 
therefore stands as follows :—For two squares the solution 
was given by Gauss in the “ Disquisitiones’’; the cases 
of four, six, and eight squares are due to Jacobi, Eisenstein, 
and Prof. Smith (see Report of the British Association for 
1865, p- 369). In these cases in which the nuaber of 
squares is even, the problem can be solved by means of 
elliptic functions, and it is not necessary to have recourse 
to the special methods of the Theory of Numbers; but 
it is not so in the case when the number of squares is 
uneven, and the question is then essentially “ arithmetical ” 
as regards its method of treatment and expression. The 
case of three squares was given by Lejeune-Dirichlet, 
and is included in Prof. Smith’s general treatment 
of ternary quadratic forms in the PAzlosophical Trans- 
actions for 1867: the enunciations for the cases of 
five squares and seven squares were given, as has been 
stated, in the Proceedings of the Royal Society for 1867. 
The demonstrations for the case of five squares have been 
communicated to the French Academy, but those for 
seven squares still remain unpublished in Prof. Smith's 
note-book. This class of questions ceases to admit of 
the same kind of solution when the number of squares 
exceeds eight, so that with the publication of the demon- 
stratioas for seven squares the solution of the whole 
problem will be complete. It will be seen that Prof. 
Smith has had a large share in this great mathematical 
victory. 
«| ’Académie était donc foadée A espérer que ce voyage de décou-~ 
vertes imposé aux concurrents & travers une des régions les plus intéressantes 
et les moins explorées de |’arithmétique produirait des résultats feconds pour 
la science. Cette attente n’a pas é1é trompée.’’ Report on the award of the 
prize, Comptes Rendus, April 2, 1883 Inthis report however no mention 
is made of the fact that these ‘‘résultats féconds’’ had been published in 
1867 
