95 
\. Helland adduces a large amount of evidence to show that the 
fjords in Norway have been formed by glacial action.—H, C. 
Vogel describes some careful experiments on the spectrum of 
aurora, which he compared with the spectra of various gases in 
Geissler tubes. He regards it as a modification of the air 
‘spectrum ; one line of the former, at least, corresponding with the 
maximum brightness of the latter, while the remaining lines 
probably appear in the spectra of atmospheric gases under 
certain conditions of temperature and pressure.—A new mode of 
Measuring rate of rotation is proposed by A, Schuller. The 
principle is briefly this: A disc divided into three sectors (black, 
red, and green), rotates on a horizontal axis ; a seconds pendulum 
fitted with a screen, in which is a vertical slit, oscillates at the 
back of it, and a ray of light passes through the slit and disc to 
a telescope through which the observer looks. The recurrence 
of particular colours observed gives a means of estimating the 
speed of rotation.—Among the remaining papers are one on a 
block of lava from the recent eruption of Vesuvius, one on com- 
pounds of thallium, and one on a new form of the Noé 
thermopile. : 
seo 
SOCIETIES AND ACADEMIES 
LoNDON 
_ Mathematical Society, Nov. ‘14.—Mr. W. Spottiswoode, 
_FE-R.S., President, in the chair.—The following gentlemen were 
‘elected as officers and members of council for the ensuing 
session :—President, Dr. Hirst, F.R.S. ; Vice-Presidents, Prof. 
Crofton, Mr. S. Roberts, and Mr. Spottiswoode ; Treasurer, 
Mr. S. Roberts; Secretaries, Mr. M. Jenkins, Mr. R. 
Tucker; other members, Prof Cayley, Prof. W. K. Clifford, 
Mr. T. Cotterill, Mr. J. W. L. Glaisher, Rev R. Harley, 
Prof. Henrici, Mr. C. W. Merrifield, Prof. H. J. S. Smith, 
Mr. J. Stirling, and Mr. J. J. Walker. Messrs. Glaisher 
and Harley were elected in the room of Dr. Sylvester and the 
Hon. J. W. Strutt. The new president having taken the chair, 
alluded in feeling terms to the loss the mathematical world and 
the Society had just experienced by the death of Dr. Clebsch, of 
Gottingen, who had been elected a foreign member in December 
Mr. Spottiswoode then read his paper, “Remarks on 
some recent Generalisations of Algebra.” It gave an analysis 
of methods used by Prof. Peirce, of Harvard, in his ‘‘ Linear 
Associative Algebra,” and by Dr. Hermann Hankel in 
his ‘‘Vorlesungen iiber die complexen Zahlen und _ ihre 
functionen,” Parti. Prof. Henrici exhibited a series of models 
‘of cubic surfaces, and pointed out the several singu- 
larities, and explained how the models were constructed. 
Prof. Clifford next read a paper ‘‘On a theorem relating to 
‘polyhedra analogous to Mr. Cotterill’s theorem on plane 
polygons,” and exhibited several illustrativemodels. The plane 
‘theorem is “for every plane polygon of 7 vertices there is a curve 
of class #— 3touching allthe diagonals ; the number of diagonals 
is such as to exactly determine this curve and no more; and 
when the curve touches the line at infinity the area of the 
polygon is zero,” The analogous theorem in space should 
‘therefore apply in the first instance to those solids whose volume 
can be expressed as the sum of tetrahedra,+having one vertex at 
_an arbitrary point of space, and the other three at three vertices 
‘of the figure; that is to say, it should apply to solids having 
‘triangular faces. For such solids the author finds that the 
analogy is very complete and exact. Defining the plane which 
‘contains three vertices and which is not a face, as a diagonal 
plane and a line joining two vertices, but not {being an edge asa 
diagonalline, he proves the following theorems :—“Forevery poly- 
hedron of # summits having only triangular faces (A faced #— 
-acron, Cayley) there is a surface of class —4, touching all the 
‘diagonal planes. This surface contains all the diagonal lines. 
‘The diagonal planes and lines are so situated, however, that the 
conditions of touching the planes and containing the lines are 
' precisely sufficient to determine a surface of class ~—4. When 
this surface touches the plane at infinity, the velume of the solid 
is zero.” Prof. Clifford then’ proceeded to apply these proposi- 
tions to polyhedra having other than triangular faces.—A paper 
“by the Hon. J. W. Strutt was, in his absence, taken as read. 
Its title was ‘‘ Investigation of the disturbance produced by 
a spherical obstacle on the waves of sound.” The problem to 
_ which chief attention is given in the paper is that of a rigid 
spherical obstacle, which is either fixed, or (more generally) so 
supported that, when disturbed from the position of equilibrium, 
it is urged back by a force proportional to the displacement. 
The mathematical solution is worked out without any limitation 
as to the size of the sphere ; but in drawing conclusions from it, 
attention is confined for the most part to the case when the 
diameter of the sphere is small compared to the length of the 
sound waves. Mr. Strutt then considers the problem of a fluid 
spherical obstacle, working it out in full for a very small sphere ; 
and afterwards he investigates anew the same problem by a very 
different analysis, not restricted to the case of a sphere or an 
abrupt variation of mechanical properties on the one hand, but 
on the other less general in requiring the variation and the 
region over which it occurs to be small. In conclusion he indi- 
cates the solution of the problem when the source of sound is at 
a finite distance from the obstacle, the primary waves being 
accordingly spherical instead of plane.—The following abstract 
of M. Hermite’s paper ‘“‘sur V’integration des fonctions circu- 
laires,” was furnished by Prof. Cayley. M. Hermite’s paper 
relates to the integral 
S (sin a, cos x) dx 
where / is any rational function of sin x, cosx. The substitution 
of ate where c= 4/ - 1, converts this into an integral of the form 
Po dz, where = is a rational function of z, viz. #, z and 
Fare each of them a rational and integral function of . The 
last mentioned integral is treated by the ordinary method of the 
decomposition of rational fractions; and the gist of the paper 
is in the transformation of the resulting expressions back from 
the new variable z to the original variable x, so as to obtain the 
required integral | f (sin x, cos x) dx, in terms of the circular 
functions of x. 
of the form i 
and integral function of z, sin x, cos v: and # x is of the form 
It is shown that the process leads to an equation 
(sin x, cos x)= x + & x, where 1 ~ is a rational 
x t p— 
~x=C+a cot vot tei cot ——+, &e, 
+4 cot 8 5, Beet 428 4+, &e. 
eee 2 ax 2 
each series, and also the number of the different series, 
being finite: so that the integration is made to depend upon 
the integrals 
[ Ml «dx and / cot —* dx=2 log sin = 
The paper contains processes for the complete determination 
of the coefficients, C, a, @,, &c. and other interesting matter. 
Meteorological Society, Nov. 20.—Dr. ‘ripe, president, in 
the chair.—On the storms experienced by the Submarine Cable 
Expedition in the Persian Gulf on November 1 and 2, 1869, by 
Mr, Latimer Clark. The first storm occurred at 9 o'clock at 
night, when the vessels of the expedition were about 130 miles 
from Bushire, and burst upon them without any preliminary 
warning, lowering the temperature by nearly 30° in a few 
minutes. It was accompanied by heavy rain and much light- 
ning and thunder, and progressed from N.W, to S.E. After 
the tempest had lasted for two hours the wind changed to a gale 
from §.E., and subsequently fell calm as before The next day 
the cable was spliced up, and paying out had scarcely re-com- 
menced, with a strong south-east wind, when notice was re- 
ceived that another violent storm from the north-west had passed 
Bushire, and was on its way down the Gulf. At 3 o'clock 
black clouds were seen rising, and at 3.52 the storm burst forth 
with the same suddenness and fury that characterised the previous 
one, Being daylight many phenomena were observed which 
were missed the night before. As the clouds approached they 
gathered into a peculiar form, resembling the cap of a large 
mushroom, extending across the heayens from one horizon to the 
other. The lower edge had a rounded and wrinkled margin, but 
was very sharply defined ; the surface was composed of innu- 
merable similar strata, as if melted pitch had been poured out 
and allowed to solidify in numerous cakes, each rather smaller 
than the one below.* Suddenly there came a profound calm, 
* This is the form of cloud mentioned by M, Poey in Nature, Vol. iv. 
No. 103. 
