— > 
23, 1870] 
Sune 
(a, . .)8(x, 3’) 5 viz., we have, according to the theory, 5= 
(10-6) +1, that is, of the form in question there are 10 compo- 
site covariants connected by 6 syzygies, and therefore equivalent 
to 10-6=4 asyzygetic covariants ; but the number of asyzygetic 
covariants being=5, there is left, according to the theory, 1 
irreducible covariant of the form in question. The fact is that 
the 6 syzygies being interconnected and equivalent to 5 inde- 
pendent syzygies only, the composite covariants are equivalent 
to 1o—5=5, the full number of the asyzygetic covariants. And 
similarly the theory as it stands gives a non-existent irreducible 
covariant (a, . .)8(x, 7)*°. The theory being thus in error, by 
reason that it omits to take account of the interconnection of the 
syzygies, there is no difficulty in conceiving that the effect is the 
introduction of an infinite series of non-existent irreducible 
coyariants, which, when the error is corrected, will disappear, 
and there will be left only a finite series of irreducible covariants. 
Although Iam not able to make this correction ina general 
manner so as to show from the theory that the number of 
the irreducible covariants is finite, and so to present the theory 
in a complete form, it nevertheless appears that the theory 
can be made to accord with the facts; and I reproduce the 
theory, as well to show that this is so as to exhibit certain new 
formulz which appear to me to place the theory in its true 
light. Iremark that although I have in my Second Memoir 
considered the question of finding the number of irreducible 
coyariants of a given degree @ in the coefficients but of any 
order whatever in the variables, the better course is to separate 
these according to their order in the variables, and so consider the 
question of finding the number of the irreducible covariants of a 
given degree @ in the coefficients, and of agiven order yw in the 
variables. This is, of course, what has to be done for the 
enumeration of the irreducible covariants of a given quantic ; 
-and what is done completely for the quadric, the cubic, and the 
quartic, and for the quintic up to the degree 6 in my Eighth 
Memoir (Phi. Trans. 1867.) The new formule exhibit this 
separation ; thus (Second Memoir, No. 49), writing a instead of 
I 
(I-a@) (1-2?) 
showing that we have irreducible covariants of the degrees 
I and 2 respectively, viz., the quadric itself, and the discri- 
minant: the new expression is ee eee showing that 
(1-ax”) (1 —a*) 
x we have for the quadric the expression 
the covariants in question are of the actual forms (a, . . }€., y)? 
and (a,. .)* respectively. Similarly for the cubic, instead of 
rae 
the expression No. 55, is we have 
(1 =a) (1-@) (1-@) r—a)’ 
1—a'x6 OF ee me. . 
— =< , exhibiting the irreducible 
(1 —a@x3) (1 — @*x*) (I —@*x*) (1-24) 
covariants of the forms (a,. . x, y)%, (a,. .)? (~, 9)?, (@. .)8 
- (x, 7)’, and (a, . .)*, connected by a syzyzy of the form (a, . .)° 
(x, y)®; and the like for quantics of a higher order. 
In the present Ninth Memoir I give the last-mentioned for- 
_mulz ; I carry on the theory of the quintic, extending the Table 
_ No. 82 of the Eighth Memoir up to the degree 8, calculating all 
_ the syzygies, and thus establishing the interconnections, in virtue 
_ of which it appears that there are really no irreducible covariants 
| of the forms (a, . .)®(«, y)4, and (a, . 2a, vy). I repro- 
duce in part Prof. Gordan’s theory so far as itapplies to the 
quintic ; and I give the expressions of such of the 23 covariants 
as are not given in my former memoirs ; these last were cal- 
culated for me by Mr. W. Barrett Davis, by the aid of a grant 
from the Donation Fund at the disposal of the Royal Society. 
_ The paragraphs of the present memoir are numbered consecu- 
‘tively with those of the former memoirs on Quantics. 
**On the Chemical Activity of Nitrates.” By Edmund J. 
_ Mills, D. Sc. 
; **On the relative Duration of the component parts of the 
Radial Sphygmograph Trace in Health.” By A. H. Garrod. 
___ Anthropological Society, May 19.—Dr. Berthold Seemann, 
_ V.P., in the chair. Mr. Henry F. Chorley read a paper on 
_ **Race in Music.” The author, after remarking on the vast 
range of the subject for treatment in the compass of a single 
_ paper, proceeded to point out the difficulties that stood in the 
_ way of determining what is and what is not truly national music, 
one great difficulty consisting in the inaccuracies of notation. 
; Notation being comparatively a modern art and the only means 
_ by which musical ideas can be transmitted, we are very much in 
the dark as to the advance made by the ancients in the art of 
music, Confining himself chiefly to the modern development of 
NATURE 
isd 
music, Mr. Chorley argued that new and original melody is far 
less common than is generally supposed. By the simple variation 
of tempo, implying some change in accentuation, a melody can 
be so entirely transformed as to lose its original character. 
Genuine, fresh, national music is, again, comparatively rare, and 
its character has always been most marked whenever intercourse 
has been the most restricted. Passing from the more limited 
subject of national music to the broader question of race-elements 
in music, the author illustrated the great distinction which exists 
between the Asiatic and the European development of the art ; 
in the former it is confined to rhythm, and seldom includes beauty 
of sound or symmetry of form. In strong contrast to the Oriental 
ideas of music were cited those of the north of Europe, viz., 
Norway, Sweden, Denmark, and Russia. In the opinion of the 
author those people take the highest place as melodists. It 
should as a fact be noted that with few exceptions those northern 
airs are in minor keys, which might be taken as an expression of, 
rather than a protest against, the gloom of the climate and scenery, 
were it not that the same characteristic largely obtains among in- 
habitants of the torrid zone. The sense of musical rhythm seems 
as distinctly marked among different peoples as varieties of physi- 
ognomy ; for instance, the Peninsular melodies are only character- 
istic when they are in triple time, the airs in common time being 
essentially mawkish and pointless, owing such individuality as they 
have to the sleepy, voluptuous delivery of the executant. On the 
other hand, the music of France lies essentially in the direction 
of squared music towards what is piquant as distinct from what 
is undulating. In treating of the subject of Race in Music the 
author could not but draw attention to a phenomenon which is 
of universal recurrence, namely, the demarcation not merely of 
race but also of sex in the art, be its stages of culture or civilisa- 
tion ever so primitive, ever so mature. The absence of musical 
inventive genius in woman is most curious and inexplicable, and 
offers another signal illustration of the contradictions and incon- 
sistencies which mark music beyond any other art. While 
women have achieved distinction and often great success in 
literature, painting, sculpture, architecture, and science, and while 
they are unsurpassed as interpreters of the drama and of the art 
of music, not a solitary female composer of originality or even 
of repute is known to the historical or critical observer. Mr. 
Dannreuther illustrated the paper by numerous examples on the 
pianoforte. 
Entomological Society, June 6,—Mr. A. R. Wallace, presi- 
dent, in the chair. Mr. F. V. Jacques, of Bristol, was elected a 
member. <A collection of insects sent to the Society by Mr. Henry 
Ansell, from Kinsembo, S.W. coast of Africa, was exhibited ; 
and another collection from Tugela, Natal, was exhibited by Mr. 
W. Warwick King. Living specimens of Azeuchus, from Venice, 
were shown by Mr. S. Stevens ; and a gynaniromorphous Zrachy- 
centrus, from Cheshunt, was exhibited by Mr. M‘Lachlan. 
Communications were made by Major Munn, on the Honey- 
Bee, by Mr. A. G. Butler, on the possible identity of A7zgynnis 
Niobe with A. Adippe, and by Mr. G. R. Crotch on the Genera 
of Coleoptera studied chronologically (part 2, from 1802 to 1821.) 
Ethnological Society, June 7.—Dr. A. Campbell, V.P., 
in the chair, R. H. Tiddeman, B.A., F.G.S., was an- 
nounced as anew member. Professor Huxley, LL.D., F.RS., 
President, read a paper ‘‘ On the chief modifications of mankind, 
and their geographical distribution.” After pointing out those 
physical characters which are of the greatest value in distinguish- 
ing the several modifications—such as colour, character of hair, 
and form of skull—the author proceeded to describe five distinct 
types of mankind: 1. The Avzsstralioid, with slender limbs, dark 
brown skin, black wavy hair, strong brow-ridges, and long skull ; 
this type is found throughout Australia among the hill tribes of 
the Dekhan in India, and formerly in the Valley of the Nile. 2. 
The Wegroid, with dark skin, black frizzled hair, and long skull ; a 
group which includes the Negroes and Bushmen of Africa, and the 
Negritos of New Guinea, Tasmania, &c. 3. The Xanthochroic, 
with fair skinand blue eyes, distributed through Iceland, Eastern 
Britain, Scandinavia, North and Central Germany, and extend- 
ing through Eastern Europe into Asia, as far as North-west 
India, and found also in the North of Africa. 4. The AZelano- 
chroic, a type with dark complexion, occupying an area situated 
between the Xanthochroic and Australioid peoples ; and 5. The 
Mongoloid, a large and somewhat ill-defined group extending 
throughout Central and Northern Asia, the two Americas, and 
Polynesia. The paper was illustrated by a large coloured map, 
showing the distribution of these five groups and their sub- 
