the yellow being the least marked. The red, green, and blue are however particularly well rendered by reflected light, and the plate shows the colours as seen when a dull light is thrown on the slit of the spectroscope, a simile which was suggested to me by Mr. Norman Lockyer. From the evidence obtained by these experiments it appears that two or three molecular groupings are sufficient to give the necessary colours, a subject which I only allude to, since the more general question of molecular groupings is being considered by others.

III. "A Tenth Memoir on Quantics." By A. CAYLEY, Sadlerian Professor of Pure Mathematics in the University of Cambridge. Received June 12, 1878.

(Abstract.)

The present memoir, which relates to the binary quintic (*)(x, y)3, has been in hand for a considerable time; the chief subject-matter was intended to be the theory of a canonical form discovered by myself, and which is briefly noticed in "Salmon's Higher Algebra," 3rd Ed (1876), pp. 217, 218; writing a, b, c, d, e, f, g . . . u, v, w, to denote the 23 covariants of the quintic, then a, b, c, d, ƒ are connected by the relation ƒ3 = - a'da2bc 4c; and the form contains these covariants thus connected together, and also e; it in fact is... (1, 0, c, f, ab3c2, a'e 2cf)(x, y)3.

But the whole plan of the memoir was changed by Sylvester's discovery of what I term the Numerical Generating Function (N.G.F.) of the covariants of the quintic, and my own subsequent establishment of the Real Generating Function (R.G.F.) of the same covariants. The effect of this was to enable me to establish for any given degree in the coefficients and order in the variables, or, as it is convenient to express it, for any given deg-order whatever, a selected system of powers and products of the covariants, say a system of "segregates ;" these are asyzygetic, that is, not connected together by any linear equation with numerical coefficients; and they are also such that every other combination of covariants of the same deg-order, say, every "congregate" of the same deg-order, can be expressed (and that, obviously, in one way only) as a linear function, with numerical coefficients, of the segregates of that deg-order. The number of congregates of a given deg-order is precisely equal to the number of the independent syzygies of the same deg-order, so that these syzygies give in effect the congregates in terms of the segregates: and the proper form in which to exhibit the syzygies is then to make each of them give a single congregate in terms of the segregates, viz., the left hand side can always be taken to be a monomial congregate aab. . ., or, to avoid fractions, a numerical multiple of such form, and the right hand

side will then be a linear function, with numerical coefficients, of the segregates of the same deg-order. Supposing such a system of syzygies obtained for a given deg-order, any covariant function (rational and integral function of covariants) is at once expressible as a linear function of the segregates of that deg-order; it is in fact only necessary to substitute therein, for every monomial congregate, its value as a linear function of the segregates. Using the word covariant in its most general sense, the general conclusion thus is that every covariant can be expressed, and that in one way only, as a linear function of segregates, or, say, in the segregate form.

Reverting to the theory of the canonical form, and attending to the relation f2 = = a3d + a2bc 4c, it thereby appears that every covariant multiplied by a power of the quintic itself, a, can be expressed, and that in one way only, as a rational and integral function of the covariants, a, b, c, d, e, f, linear as regards f; say, every covariant, multiplied by a power of a, can be expressed, and that in one way only, in the "standard" form. As an illustration, we may take a2h= 6acd4beef. Conversely, an expression of the standard form, that is, a rational and integral function of a, b, c, d, e, f, linear as regards f, not explicitly divisible by a, may very well be really divisible by a power of a (the expression of the quotient, of course containing one or more of the higher covariants, g, h, &c.), and we say that, in this case, the expression is "divisible," and has for its "divided" form the quotient expressed as a rational and integral function of covariants. Observe that, in general, the divided form is not perfectly definite, only becoming so when expressed in the before-mentioned segregate form, and that this further reduction ought to be made. There is occasion, however, to consider these divided forms, whether or not thus further reduced, and moreover it sometimes happens that the form presents itself or can be obtained with integer numerical coefficients, while the coefficients of the corresponding segregate form are fractional.

The canonical form is peculiarly convenient for obtaining the expressions of the several derivatives (Gordan's "Uebereinanderschiebungen "), (a, b)', (a, b)2, &c. (or, as I propose to write them, abl, ab2, &c.), which can be formed with two covariants the same or different, as rational and integral functions of the several covariants. It will be recollected that, in Gordan's theory, these derivatives are used in order to establish the system of the 23 covariants, but it seems preferable to have the system of covariants and, by means of them, to obtain the theory of the derivatives.

I mention, at the end of the memoir, two expressions (one or both of them due to Sylvester) for the N.G.F. of a binary sextic.

The several points above adverted to are considered in the memoir; the paragraphs are numbered consecutively with those of the former memoirs upon quantics.

IV. "Osteology of Polyodon folium." By T. W. BRIDGE, B.A., Scholar of Trinity College, and Demonstrator of Comparative Anatomy in the University of Cambridge. Communicated by Professor NEWTON, F.R.S. Received May 29,

1878.

(Abstract.)

On a previous occasion* the author described the cranial osteology of one of the more specialized genera of existing Ganoids, and the paper now laid before the Society is an attempt to add to the relatively scanty literature of this department of comparative anatomy, by giving in detail an account of the skeletal structures of one of the most generalized members of the group. The following are among the more important of the conclusions which the facts elucidated appear to warrant.

A comparison of Polyodon with its nearest living ally, the Sturgeon, demonstrates the close relationship that exists between the two forms as regards their skeletal anatomy. In so far as Acipenser differs from Polyodon, it approaches the Teleostean type, nor is the latter genus without indications of having undergone a degree of specialization in the same direction. But not the least important characteristic of Polyodon is its possession of a remarkable combination of structures, usually regarded as being specially distinctive of the Plagiostome Elasmobranchii. The existence of such primitive characters, which are also present in Acipenser, though to a less extent, suggests the desirability of giving to such facts adequate expression in any scheme that may be proposed for the classification of the Ganoids. After giving a resume of the views held by various zoologists as to the position of these genera in their order, it is suggested that J. Müller's two suborders (Chondrostei and Holostei) should be retained, though it seems advantageous to remodel the definitions on which they were based. The various families of recent Ganoidei may be arranged in two suborders of approximately equal morphological value, which are mainly differentiated by the structure of their upper jaws.

a. G. Selachoidei.-Pterygoid processes united in a medio-ventral symphysis beneath the basis cranii; notochord persistent and unsegmented; spiracles and mandibular branchiæ present.

Gen. Polyodon, Acipenser, Scaphirhynchus, Chondrosteus.

B. G. Teleosteoidei.-Pterygoid processes not united with each other but are connected directly, or indirectly by the intervention of a palatine bone, with the prefrontal region of the skull; vertebral column generally ossified; mandibular gills abortive.

Gen. Amia, Lepidosteus, Polypterus.

* "Journal of Anatomy and Physiology," vol. xi.

If tested by the structure of their upper jaws, the Crossopterygidæ, Lepidosteidæ, and the Palæoniscidae should be referred to the Teleosteoidei, while the Acanthodida would probably belong to the other sub-order. Nevertheless, the distinctness of the two groups is materially lessened by the existence of many annectent fossil forms. Acipenser and Chondrosteus in the one division and Palæoniscus in the other, partially bridge over the gap which exists between the two when the recent forms only are considered. The Placodermi and the Cephalaspidæ must still be referred to as being "incertæ sedis."

A comparison of the skull of Polyodon with the Amphibian skull leads to interesting results. Perhaps the most remarkable feature in which Polyodon resembles the Anura, is in the possession of a forwardly directed "orbitar process" associated with a suspensorium so much inclined backwards that the gape is extended even beyond the posterior limits of the skull.

The condition of the "orbitar process," as an apparently functionless rudiment in Polyodon, does not throw any light on its primitive origin, but its position and relations in the adult Lamprey, and its transitory condition in the embryo Anura, suggest that originally it may have acted as an anterior suspensor to the much inclined mandibular pier of animals possessing a suctorial mouth, prior to the adaptation of the pterygo-quadrate arcade to that purpose. These facts, together with the rotation of the quadrate cartilage, which we may infer to have taken place from the direction and relations of the "orbitar process," are indications of the existence of a close parallelism between the developmental history of the cranium in the embryo Polyodon, and in such otherwise dissimilar Anurous Amphibia as Dactylethra capensis, Bufo ornatus, and Rana temporaria. The fenestration of the roof of the periotic capsule which exists in Polyodon seems to correspond to the primitive auditory involution which persists in Siren lacertina, and, as in the latter, it is situated to the outer side of the arch of the posterior vertical semicircular canal, and not to its inner or mesial side as is the case with the Selachians.

The co-existence in Polyodon of so remarkable a combination of Amphibian and Selachian features suggests an enquiry into the phylogenetic relationship of the Ganoids, the Amphibia, and the Elasmobranchs, or, in other words, suggests the question, is the ancestral stem of the Ganoidei more closely related to that of the Amphibia, or to that of the Elasmobranchs? An analysis of the structural features common to any two of these primary groups seems to affirm the monophyletic origin of the two first-mentioned. On this assumption the relation of the three groups may be roughly and tentatively expressed as follows:

It seems not improbable that a primitive ancestral stock (x) very

early differentiated into the two groups of Apneumatocola and Pneumatocola, the former being the root-stock of the modern Elasmobranchii, while the latter, by acquiring rudimentary and more or less functional lungs, became the primitive double-breathers from which have been derived the Ganoidei and the Amphibia. From the primi. tive Ganoidei were derived the Teleosteoid Ganoids and eventually the Teleostei also, their originally complex swim-bladders becoming gradually devoted to other functions, while the Selachoidei may be regarded as the but little modified descendants of the original progenitors of the order. The close correspondence that exists between Polyodon and the Selachii is not incompatible with these views, but may be the result of the persistence in both of structures originally possessed by their primitive ancestor. Two facts in the cranial anatomy of Polyodon are not easy to explain, viz., the formation of the upper jaw and the existence of the "orbitar process." The union of the pterygoid processes in a median symphysis may have been the primitive condition of the jaws in the ancestral form (x), but that while persistent in Polyodon and in the Selachii, it was superseded by a different arrangement, viz., the union of the pterygoid processes with retral palatine outgrowths in most Ganoidei, and in all Teleostei and Amphibia. Neither is it easy to account for the retention of the " orbitar process." It may have been an adaptive modification correlated with a suctorial mouth in the larval or adult forms of those Ganoids that were first differentiated from the Amphibian stem, and independently developed; or it may have been possessed by, and similarly functioned in, the primitive Pneumatocola, but has become obsolete in all their descendants, except Polyodon and the Anura. Thus it would appear that the Polyodontidae constitute a remarkably central group. They retain not a few of the characters which we may assume to have belonged to the primitive stock out of which were evolved the three most important groups of Ichthyopsida, combined, however, with a certain amount of specialization; nor are they altogether without indications of retrogression.

V. "On Astrophiura permira, an Echinoderm-form intermediate between Ophiuroidea and Asteroidea." By W. PERCY SLADEN, F.L.S., F.G.S. Communicated by Professor DUNCAN, F.R.S. Received June 18, 1878.

(Abstract.)

The following peculiarities of structure presented by the anatomy

of the echinoderm above described are noteworthy:

1. The combination of ophiuroid disk- and arm-structure within a pentagonal asteroid form of body.