At the Observatories of the Cape, of Colaba at Bombay, and of Rome, it was only necessary to enter the approximate local mean time under the heading, "Chronometer," as the declination at these stations was measured from the standard marks. The bearing of these magnetic marks is respectively 0° 31′ 51′′, 0° 26′ 26′′, and 21° 26′ 54′′ E. of the Astronomical North Point. At the remaining stations the sun's azimuth was computed for the hour angle of the sun given by the chronometer, except at the college of Bombay, where it was found more convenient to observe the stars Aldebaran and Capella, instead of the sun. The observations marked with an asterisk were taken with the needle of the Cape Observatory. The computed altitudes of the sun for the three observations at Moncalieri, were 63° 3′ 33′′ 60, 63° 48′ 57′′ 69, and 64° 29′ 17" 78. This difference of altitude being so slight makes the correction for the perpendicularity of the mirror much less reliable than it might otherwise have been.

The declination observations at the Cape of Good Hope date back as far as 1600, and the mean yearly amount of the secular variation for the first 240 years was found to be + 7'66 W. It then diminished, and was +0'49 from 1841 to 1846, +1'41 from 1843 to 1848, and 2'16 from 1846 to 1850. The renewed acceleration in the annual variation seems to have lasted but for a short period, as a comparison of the mean value for 1874, with 29° 14' 62, the mean for 1848, gives only 1'68 as the annual change.

From observations of this magnetic element at Bombay in 1845, 1856, 1867, and 1875, we notice a gradual easterly motion of the needle, it being at the above dates 0° 13′ W., 0° 19′ E., 0° 42′ E., and 0° 56' E., shewing a decreasing yearly change of 2'91, 2'09, and 1'·75. The amount of the mean annual variation given by the series of monthly observations at the Colaba Observatory from 1867 to 1873 is 1'77, which may be considered identical with the above.

At Aden the heat of the sun was so great that it was imprudent to remain long near the theodolite whilst taking the solar observation, the declination was therefore obtained under very disadvantageous circumstances, and consequently less weight can be attached to the result. This is apparent also from the abnormal increase in the secular acceleration. Thus the declination which was 5° 2′ W. in 1834, became 4° 15' W. in 1857, the annual variation being, therefore, -2'04; whilst in 1875 the angle observed was 2° 19′ 39′′ W., which increases the yearly change to -6'41.

The declination in 1875 at Malta, combined with the value 15° 20′ W., obtained in 1834, gives a yearly change of 4'76; and the angle 13° 49′ measured at Rome in 1852-5, along with the value found in 1875, makes the annual variation -5'95. For the remaining stations I am unacquainted with any published values of this element of the earth's magnetic force, but the careful series of

measures taken at Moncalieri, by P. Denza, will most probably appear shortly in the publications of that Observatory.

In conclusion I will subjoin in a single table the mean results for all the elements of terrestrial magnetism at the different stations.

II. "On the Limits to the Order and Degree of the Fundamental Invariants of Binary Quantics." By J. J. SYLVESTER, M.A., LL.D., F.R.S., Professor in the Johns Hopkins University, Baltimore, U.S. Received December 26, 1877.

The developments which I have recently given to Professor Cayley's second method of dealing with invariants (the first method being that which has been exclusively used by Professor Gordan), has led me through the theory of the Canonical Generating Fraction to the following results, showing that the degree and order of the fundamental invariants and covariants to a quantic or system of quantics are subject to algebraical limits of a very simple kind, and I think it right that these results should not be withheld from the knowlege of those who are pursuing another and, as it seems to me, much more arduous and less promising direction of inquiry into the same subject.

By order I mean the dimensions of a derived form in the coefficients of its primitive (Clebsch and Gordan's grad), and by degree the dimensions in the variables (Clebsch and Gordan's ordnung).

First as to degree.

If there be a system of n, n' n".. odd degreed quantics and , ', .. &c., even ones, then (with the exception of the case when the system reduces to a single linear function or a single quadratic) the degree of

any irreducible covariant to the system has for a superior limit

Thus, ex. gr., where there is but one quantic, the limit is

»2-4, according as the degree is n odd or v even.

2

Secondly, as to order.

n2-3

or

As the expressions become somewhat complicated when there are several quantics, I shall confine myself to a statement applicable to a single quantic, distinguishing between the three cases when n (its degree) is evenly even, oddly even, and odd.

A. When n contains 4, the superior limits for the order of the inva(n+1)(n−4) riants and covariants respectively are for the former 2

B. When n is even, but not divisible by 4, and is greater than 2,

C. When n is any odd number greater than 3, the order of the

3

invariants has for its limit

(n+1) (n-3), and when it is any odd.

2

number greater than unity, the order of the covariants has for its

Further investigations will, I have good reason to believe, lead to considerably lower limits than those given for cases B and C.

Although morally certain the three formulæ A, B, C cannot be considered at present apodictically established, the formula respecting the limit to degree may, I believe, be regarded as admitting of a complete demonstration. There exists, however, a superior limit to the orders of the fundamental invariants or covariants, which may be regarded as subject to direct demonstration even in our present state of knowledge; this when n is even is n2 - 2n 3 for invariants, and n2n 4 for covariants; and when n is odd, the corresponding limits are 2n2-3n-5 for invariants, and 2n2-2n-5 for covariants. But I have no moral doubt whatever of the validity of the formulæ B and C as they stand, and next to none of the validity of formula A.

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III. "On the Structure and Development of the Skull in the Common Snake (Tropidonotus natrix). By W. K. PARKER, F.R.S. Received October 15, 1877.

(Abstract.)

For several years past I have taken every opportunity to work at the skull of the common snake, but sufficient materials for completing a memoir upon it have not turned up until lately.

Last year, however, at the request of my friend Mr. P. H. Carpenter, Dr. Max Braun, of Würzburg, kindly sent me about fifty early embryos of reptiles; these were of four kinds, namely, of the common snake, the blind worm, the nimble lizard, and the gecko.

These invaluable embryos, added to what I have for many years been collecting, will enable me to bring before the Society a yearly tribute of a paper on the skull of these instructive types.

Lying at the very base of the gill-less Vertebrata, and possessing a skull at once the simplest, and yet the most curiously specialized, the snake is a type well worth careful study. I have found it so in my own division of work.

My guide in this piece of work has been Rathke, whose observations on the early stages of the skull appeared first, in translation, in this country, in Professor Huxley's "Croonian Lecture."

For many years past I have known that the key to the meaning of the skull in the whole series of the "Sauropsida "-reptiles and birds in one huge group was to be found in that of the serpent. I have freely used it as such, and my nomenclature of the parts of the growing bird's skull is based on that of the snake's, although, until now, I have not been able to publish more than a mere abstract upon its characters.

For convenience' sake, I have divided my subject into seven stages, the first of these being illustrated by embryos with a very delicate vesicular head, bent down upon the neck, and whose entire length, supposing them to be uncoiled, was barely three-quarters of an inch in length. The last stage is the adult, whose skull, once interpreted, will greatly help in the interpretation of all the skulls that rise above and around it.

As to the finished skull, it is easy to judge beforehand that the cranial part must be a very solid and relatively small box, and that the facial part must be free and elastic to the utmost degree.

All the morphological specializations that take place in the head of the embryo steadily lead to this result; but the superstructure is marvellously unlike the foundations that were at first laid.

As to the general embryological study of the snake's embryo, it may be remarked that it is almost exactly like that of the bird's.