224. 
NARORE 
[Dee. 23, 1869 
and the foci of the plane. Ifwe take any third point in the plane, 
the angles which its focal radii vectores make with the line joining 
the foci are equal to the angles which the traces of the corre- 
sponding planes upon the cyclic planes make with the line of 
intersection of those two planes. ‘These theorems suppose only 
that in the two correlative figures the plane at an infinite distance 
in either figure answers to a point at a finite distance in the other. 
Mr. Tucker (Hon. Sec.) read a proof (by Mr. M. W. Crofton, 
F.R.S ) of Gauss’ Theorems and Napier’s Analogies. ‘The proof,a 
purely geometrical one, was extremely neat and simple. The 
perpendicular bisector of the base is produced to meet the ex- 
ternal bisector of the vertical angle, and from the point of inter- 
section (P) arcs are drawn perpendicular to the sides containing 
the said vertical angle. This point (Pp) is also connectcd by 
arcs with the extremities of the base ; the results readily follow 
from the equality of certain triangles. —Mr. S. Roberts, M A., 
gave an account of a short paper On the Order of the Discrimi- 
nant of a Ternary Form. The main theorems, the author states, 
have been geometrically obtained by, and are due to, Professor 
Cremona(Mr. Roberts’ paper contains an analytical proof of them), 
and relate to the influence of common multiple points on the 
number of double points of a pencil or involution of curves. The 
method employed was applied to the determination of the dis- 
criminant of a ternary form when certain of the terms are want- 
ing, viz. the form (x, 3’) P(x, 2)1. The communication also 
had reference to a paper on Discriminants, by Dr. Henrici, pub- 
lished in the Society’s Proceedings, in which the result had been 
arrived at zdirect/y. and Mr. Roberts’ aim was to clear up an 
apparent discrepancy in the results obtained by Professor Cre- 
monaand Dr. Henrici. The latter gentleman joined in a discus- 
sion on the subject.—The President gave an account of his 
investigations on the cen ro-surface of an ellipsoid (locus of the 
centres of curvature of the ellipsoid). The surface has been 
studied by Dr. Salmon, and also by Professor Clebsch, but in 
particular the theory of the nodal curve on the surface admits of 
further development. The principal sections of the surface (as 
is known) consist each of them of an ellipse counting three times, 
and of an evolute of an ellipse : the evolute and ellipse have four 
contacts (twofold intersections) and four simple intersections, 
but the contacts and intersections respectively are in the different 
sections real or imaginary. The form of the principal sections 
then is: a real contact at Pin the plane of «xz, and a real in- 
tersection at Q in the plane of «xy, and thus there are an exte- 
rior and an interior sheet, but (instead of meeting in a conical 
point, as in the wave surface) these intersect in a nodal curve 
qr. The curve has a cusp at Q, and a node at Pp, thus ex- 
tending beyond p, but from that point is acnodal, or without 
any real branch of the surface passing through it. Several 
simple relations were established, but the reductions were of 
some complexity. 
Syro-Egyptian Society, December 7th.—W. H. Black, 
Esq. in the chair. Mr. Bonomi read a paper on the defacement 
of the name and figure of the god Amon on all Egyptian tem- 
ples, obelisks, and statues, during the reign of the successor of 
Amnnohph III., and the subsequent restoration of both during 
the reign of Rameses II. Mr. Bonomi conjectures that the 
amount of skilled Jabour and expenses of scaffolding necessary 
to effect these changes prove that they were considered of great 
importance in a religious point of view; and he stated that 
there was scarcely a public or private collection of Egyptian anti- 
quities in Europe that could not furnish examples of it. The 
beautifvl drawings and photographs exhibited by Mr. Bonomi, 
to illustrate the subject, enhanced the interest of his able paper. 
Institution of Civil Engineers, December 14.—Mr. Chas. 
Hutton Gregory, president, in the chair. The paper on Ocean 
Steam Navigation, with a view to its future development, by 
Mr John Grantham, M. Inst. C.E., partly read at the meeting 
on the 7th inst., was concluded; and a brief abstract of the whole 
is now given. ‘The author contended that steam-ships could be 
employed more extensively on routes partially occupied by them, 
and on others where regular steam-lines had not yet been estab- 
lished. Rapid and regular voyages both for passengers and 
goods were now fully appreciated, while the greatly-increasing 
intercourse of all nations furnished freights which would support 
lines of expensive steam-vessels. He traced the rise of ocean 
steam navigation, and showed that the route from Liverpool to 
New York was the principal field on which it was first fully 
developed. He described the efforts made by the Americans to 
maintain by steam the prestige so long secured by their sailing | 
ships; gave the reason for the great change that had taken place; 
and stated that not one American steamer was now running be- 
tween Europe and America. Some of the causes of this were 
to be found in the fact, that iron ships, worked by the screw 
propeller, could alone be employed successfully, and that such 
ships in America were too expensive, both in their construction 
and in their working, to enable them to compete with English 
vessels. The form of, and various improvements in, the boilers 
and the engines were described, showing that a much higher 
pressure of steam was now employed, that the expansive system 
and surface condensation were at present considered essential to 
success in economising fuel, and that the amount of coals con- 
sumed had in the best vessels been reduced to 24]b. per indi- 
cated horse-power per hour, but it was anticipated that a 
reduction to 2lb. might soon be attained. A map on Mercator’s 
projection was exhibited, indicating the principal ocean routes in 
connection with the trades between Great Britain and the rest 
of the world; and the improved system by great circle sailing, 
as recommended by Mr. Towson, of Liverpool, was described. 
A table was also exhibited of the relative distances between 
London and Liverpool, and the various ports shown on the map, 
both by long sea, and by the Suez Canal and the Pacific Railroad ; 
from which it appeared that, as regarded the Northern Hemi- 
sphere, a great saving of distance and time would be effected. 
Yhe number of ocean steam-ships now working in connection 
with this country was stated to be 364. The performances of 
the best ships of various companies were then alluded to, and 
the result showed that on the North American lines the highest 
average rate of speed was maintained, but by a large expenditure 
of fuel; that the Pacific and Colonial Companies’ ships gave 
excellent results, as regarded economy of fuel; and that some 
new vessels, lately built for the Royal West India Mail Com- 
pany, seemed to promise the best performances with respect to 
speed and economy combined. A table taken from the Board of 
Trade Returns for 1868 showed that, with North America, the 
tonnage of steam-ships nearly equalled that of sailing ships ; but 
in the Mediterrancan trade steam-ships largely exceeded sailing 
ships. On the other hand, in the regular trades with India, 
China, and Australia, steam tonnage, by long sea, comprised only 
about 1 per cent. of the whole. A calculation was then made to 
show what might be expected if the trade with the East was in 
future carried through the Suez Canal, and of the number of 
large steam-fleets which would be required to work it. Some 
facts were also recorded relative to the effect of the Pacific 
Railroad, and the probability of letters and passengers from 
China, Japan, and Western Australia going by that route. It 
was shown that several days’ saving in time would be effected. 
The author considered that the voyage to Melbourne cculd be 
best performed by long sea, as there would be no saving either in 
distance or in time by way of Suez. The paper held out 
great prospects of advantage to England and to British ship- 
builders, from the immense changes that were apparently about 
to take place. 
Anthropological Society, December 14.—Dr. Charnock, 
V.P., in the chair. Mr. Wake read a paper on the Race 
Affinities of the Madecasses. The agreement of the /Yovas with 
the other inhabitants of Madagascar in language and customs 
forbids us to refer the former to a Malayan origin. Moreover, 
the division into dark and light tribes is found in the Malayan 
Archipelago, and also in South Africa. Comparison of physical 
and mental characters, and of customs and superstitions, shows 
that the dark Madecasses, the Kafirs, and the Papuans, all 
belong to the same race. By a similar comparison, an analogous 
affinity between the Hovas, the Hottentots, and the Malays (as 
representative of whom the paper took the Siamese), can be 
established. The South African relationship of the Madecasses 
is supported by the verbal and grammatical affinity of the 
Malagasy to the Kafir and Hottentot dialects, which are shown 
to be related between themselves, and also to the Polynesian 
dialects. The arrangement of peoples on the African area is 
opposed to the idea of a continental origin of the Madecasses, 
while their numerous African affinities prevent their being traced 
to a Malayan source. The Madecasses are more really azfo- 
chthonous than any other race except the aborigines of Australia, 
and probably Madagascar was connected with both the African 
Continent and the Malay Archipelago when it was first inhabited 
by man. ‘The inhabitants of Madagascar possess the domestic 
ox, sheep, and fowl, and are skilled in the smelting and working 
of iron. This island was probably (according to the author) 
the seat of man’s primitive civilisation, 
