August 19, 1886] 
NATURE 
365 
admit of being explained by any previous theory. It only 
remains to add that, if true, the present theory ought to 
- admitof experimental verification. Let well-marked natural 
varieties of plants growing on the same area be system- 
atically tested with regard to their relative deg7ees of 
fertility, first within themselves, and next towards one 
another: let these experiments be made in successive 
years over a number of natural varieties, by carefully- 
conducted artificial fertilisation, and by counting the 
seeds and tabulating the results. In this. way experi- 
mental evidence would probably be obtained of degrees 
of sterility between even slight though constant varieties 
growing on the same areas; and, if so, such evidence 
would serve as further proof of the present theory. But 
experiments of this kind, in order to be satisfactory, 
ought to be conducted by a number of observers in 
different geographical areas ; and my object in publishing 
so lengthy an abstract of my views in this periodical is 
that of inducing naturalists in other parts of the world to 
co-operate with me in carrying out this research. The 
paper itself, which furnishes fuller particulars as to the 
way in which such experiments should be carried out, is 
published in a separate form by the Linnean Society. 
THE WOODEND COLLIERY EXPLOSION 
OUI sexcuse S'accuse will occur to the minds of many 
— who have followed the details of the disastrous 
explosion which took place at Woodend or Bedford 
Colliery on Friday last. We read in the 7Zzmes of the 
16th inst.:—‘ The Four-foot or Crombonke Mine is a 
very dusty one, and it is considered that at the Woodend 
pit the dust has increased the extent of the damage.” 
“But to water the mine, as suggested by the Commission, 
would here be a very difficult operation, because the floor 
of the mine consists of a species of fire-clay which, as it 
absorbs the water, causes a lifting of the ground, and so 
prevents mining operations being conducted.” Inasmuch, 
however, as the floor of perhaps ninety-nine out of every 
hundred mines consists of the same kind of material, the 
same argument against watering would hold equally good 
in most cases, and, if it is allowed to pass, this recom- 
mendation of the Commissioners is likely to come to 
nothing. It has been pointed out more than once in 
NaTurRE that the amount of water required to lay the 
dust is very small—far less than would be necessary to 
materially affect the floor of a mine in the manner sug- 
gested, and it would perhaps be wiser to try the effect in 
the first place and judge by results rather than to meet the 
proposition with a simple 707 fossumus. We speak thus 
plainly here, because many of the witnesses who gave evi- 
dence before the Commissioners brought forward the very 
same argument with the same degree of plausibility,and we 
have reason to believe without having put the matter to a 
practical test. Many of those who now water regularly, 
for the express purpose of laying the dust on floors con- 
sisting of fire-clay, admit that the water produces no 
appreciable difference when properly and carefully distri- 
buted. 
The bursting of the gauze of a safety-lamp, described 
by one of the survivors, is so contrary to all reason and 
experience that it cannot be accepted as an explanation 
of the origin of the explosion. Hundreds, if not thousands, 
of safety lamps are placed in explosive gas every day 
when the mines are being tested for the presence of fire- 
damp, and yet no parallel case has ever been recorded. 
Under these circumstances we prefer to attribute it to 
some other still unknown cause. We have yet to learn 
whether shots were fired in the mine, and if so we have 
probably not far to look for the explanation. 
Up to the present all we know with certainty is that 
the mine produced very little gas, that it was dry and 
dusty, and that the explosion was violent but not univer- 
sal. It would be most interesting, as well as instructive, 
i 
to ascertain whether any natural local dampness curtailed 
its extent ; but as this is a feature that has not hitherto 
attracted or received much attention, we are not sanguine 
that it will be carefully inquired into in the present case. 
We shall, however, watch the future course of the inquiry, 
and perhaps again comment upon it for the benefit of our 
readers. W. G. 
ON THE DIFFERENTIAL EQUATION TO A 
CURVE OF ANY ORDER 
O Mr. Samuel Roberts (see Reprint of Educational 
Times, vol. x. p. 47) is due the credit of having 
been the first to show that a direct method of elimination 
properly conducted leads to the differential equation for 
a cubic curve: but he has not attempted to obtain the 
general formula fora curve of any order. By aid of a 
very simple idea explained in a paper intended to appear 
in the Comptes rendus of the Institute, I find without 
calculation the general form of this equation. The 
eft-hand member of it may be conveniently termed 
the differential c7ztevzon to the curve. One single matrix 
will then serve to express the criteria for all curves 
whose order does not exceed any prescribed number. 
For instance, suppose we wish to have the criteria for the 
orders I, 2, 3, 4:— 
Let 7 4 be used in general to denote the coefficient of 
jm in (1 ye 2 ye + yt. .) : 
1.2 L237 easy 
Write down the matrix— 
Bat 3° Bi 2) 42h Gge2 gasp oul) | Su zioy Sime ou 
Ber. al, 42 ene se2) Seam Gun (OS2PaOraummcea 
A 5 oh 52, O"Nl ) O22. Gea ee ayes act 
ier. G6: 6.2) 70 72a CS OLS OL. 
Og Rie Pa Ey Ie SE GP. GP YOR 
7 SE RS GRO POT, NG) OP} sO 
Sep Ost Oh2) SHOT TO;2) se TOs Ua Teme Lomeli mei 
Out. 10st) |LOt2e bat alee oie ast coo Ros meet 
Wer poeie piece gee pe Vibe eit Welty seal) heh 
Migr tr2°1 | 12:2 13" Es; 2ye lg: sme4 cme lALoeO rd omg, 
The determinant of the entire matrix, which is of the 
tenth order, is the criterion for a quartic curve. The 
determinant of the minor of the sixth order, comprised 
within the first six lines and columns is the criterion for 
acubic. The determinant of the third order, comprised 
within the first three lines and columns (subject to a 
remark about to be made) will furnish the criterion for a 
conic, and the apex of the matrix is the criterion for 
the straight line. By adding on five more lines and 
columns, according to an obvious law, the matrix may be 
extended so as to give the criterion for a quintic; then 
six more lines and columns a sextic, and so on as far 
as may be required. 
The remark to be made concerning the determinant of 
the third order referred to is that it contains the irrelevant 
" 
Bn oh ore 2 
factor 2°I, ¢.c. —,so that the criterion for a conic (Monge’s) 
2 
is this determinant divested of such factor. It is certazw 
that the next determinant is indecomposable, and is 
therefore the criterion for a cubic. There is no reason 
that I know of to suppose that any other determinant 
except that one which corresponds to the conic, is decom- 
posable into factors. If this is made out, then, observing 
that the single term which is the criterion for the right 
line is indecomposable, we have another example of what 
may be called, in Babbage’s words, a miraculous excep- 
tion to a general law. 
A well-known similar case of such miraculous excep- 
tion I had occasion many years ago to notice in connec- 
tion with the criteria for determining the number of 
real and imaginary roots in an algebraical equation. 
Such criteria may, with one single exception, be expressed 
