is pupil of pupil of does not exist. The rule of combination is that two relations give a new relation, that of antecedent of first to consequent of second, if consequent of first is antecedent of second; otherwise they give zero.
   Using the same rule of multiplication consider the expressions, -never mind their meaning,-"

it will be found that the multiplication table is

precisely that of the quaternion units.
   Is all this mathematics? Has the idea of quantity for a moment entered in? The example is from Charles Pierce's Logic of Relatives. He has among other algebras expressed all of the two hundred odd of his father's "Linear Associative Algebra" in this notation.
   Take another example, this time from the theory of groups.

   Let (lh) denote the operation that changes love to hate and hate to love, while (wp) similarly interchanges wealth and penury.
   Then (lh)² = 1, i. e., leaves all as it was.
   Likewise (wp)² = 1.
   While (lh) (wp) gives both transformations at once.
   Call (lh), (wp), (1h) (wp),

   a,   b, and  c respectively. The multiplication table is

   The similarity to the quaternion table is manifest. In fact, the quaternion units are identity and three quarter rotations, while here we could take for units identity and three half-rotations.
   Any meanings whatsoever may be given to our symbols that are consistent with the purely formal laws of combination. It is not the subject-matter, but the character of the reasoning and the method of carrying it on, that makes the science rather ab-

stract. The reasoning is deductive, rather intricate, and generally carried on by an elaborate symbolism. Wherever this is so, whether in physics, chemistry, or biology, economics, logic, or philosophy, we recognize it as mathematics and we know that only the mathematical mind can successfully grapple with it.
   I plead, then, that all who have, in any degree, mathematical power should, no matter what their chosen line of work, develop that power. At any time an occasion demanding the use of that power is liable to arise. I would that a large proportion of scientific men, especially, could have what Darwin has called their "sixth sense" developed. I would, too, that all mathematicians could take at least a master's course in some non-mathematical science. It seems to me that no one science can so well serve to co-ordinate and, as it were, bind together all of the sciences as that queen of them all, mathematics.

A FAMILY OF QUARTIC SURFACES.

ROBERT E. MORITZ.

   The principal surface in this family was discovered during an attempt to construct the locus of a point so moving that the sum or difference of its distances from two intersecting straight lines is constant.
   Setting up the equation of condition, using rectangular Cartesian co-ordinates, taking the line bisecting the angle between the directrices for the x-axis, a line perpendicular to their plane at their point of intersection for the z-axis, calling 2k the sum or difference of the distances of the running point to the directrices, and 2ø the angle between the directrices, we obtain, after proper reductions,

   If now we put k² / sin²ø = a²,  k² / cos²ø = b²,  and sin²ø cos²ø / k² = c², the equation assumes the symmetrical form

   This quartic surface possesses the following remarkable features:
   (1.) Two of the parallel systems of sections of this surface are coaxal systems of conics.
   (2.) The sections parallel to the third co-ordinate plane are curves of the fourth degree, having in general four infinite branches, and, near the principal section, an oval besides. The principal section consists of two pairs of parallel lines.
   (3.) The locus of the asymptotes to either system of coaxal conics forms a companion surface which is also of the fourth order. These two companion surfaces intersect in two plane curves.

   (4.) Each of the companion surfaces contains, among all the possible systems of parallel sections, one system of coaxal hyperbolus. The locus of the asymptotes of these hyperbolas form two hyperbolic paraboloids, intersecting each other in two straight lines.
   (5.) These two hyperbolic paraboloids have each a pair of asymptotic surfaces, whose equation is xy = 0

   Features (1), (2), and (3) are represented in Plate VII.

   If now we consider a², b², and c² as arbitrary constants, capable of assuming all values from + ° through 0 to - ° we get seven other surfaces, six of which are real, one imaginary, but all closely related to the principal surface. The remarkable relations existing between corresponding cross-sections of each pair of surfaces is brought out in the following exhibit of results. The following abbreviations are used: E. for ellipses, L. for lines, I. E. for imaginary ellipses, H. for hyperbolas, and C. H. for hyperbolas lying along the z-axis.

   The study of the form and curvatures of these surfaces leads to the following results:
   (1.) Surfaces I, II, VII, and VIII have regions (if both elliptic

and hyperbolic curvature and these regions are separated by lines of parabolic curvature.
   (2.) Surfaces III and V have hyperbolic curvature only.
   (3.) Surfaces IV and VI have elliptic curvature only.
   The paper, of which this is an abstract, is accompanied by ten figures and eight plates, representing the several surfaces in parallel perspective. The paper will be published in full elsewhere.

   Hastings College, Hastings, Nebr.,
      February, 1897.

OSCAR VAN PELT STOUT.

   (Printed in full in the Transactions of the Nebraska Engineering Society, Vol. I, No. 1, pp. 13-16.)

NOTES ON PHYLLOPOD CRUSTACEA.

H. ANDERSON LAFLER AND A. S. PEARSE.

   It is greatly to be regretted that so interesting a sub-order as the Phyllopoda, a group characteristic of the plains region, one genus being peculiar to it, has been so completely neglected by our western naturalists. These creatures possess very singular means of adaptation to changed environment and the greatest vitality of species, although weak and delicate as individuals. Their method of reproduction is so bizarre as to excite the greatest interest in the student. Their broad, leaf-like feet are the characteristics from which the sub-order derives its name, Phyllopoda. The carapace of the higher genera consists of a broad, thin plate, which covers the anterior portion of the body. In the lower forms it is bent downward, forming two valves similar in appearance to those of some small mollusks. These enclose the entire body.
   Our Phyllopods are found in puddles such as are left after rains, in buffalo wallows, in slight hollows made by excavations for railway embankments, in draws which dry up during the summer months, and in places of similar nature. The eggs after being carried for a time in the egg sacs, are allowed to drop to the bottom of the puddles. The water evaporates during the summer and leaves the eggs in the dry mud exposed to the heat of summer and the cold of winter until the hollows fill again and conditions are favorable to their development. The eggs then hatch out and the cycle of life is again begun.

   At De Witt, Nebr., where most of our specimens were taken, Apus lucasanus was one of the most common species. It was first

observed on June 16, 1895, occurring abundantly in pools by the side of railway tracks. It was also abundant in a draw about one mile north of that place. Some specimens were secured and placed in a large jar, but they lived only a few hours. One or two of the more vigorous individuals were observed sucking the blood of their weaker companions. The bodies of the latter were pale and almost devoid of blood, while those of the former were gorged and of a dark red color. The same thing was noted at a later date of two specimens in a pool. This fact is of peculiar interest, as Dr. Merrill, of the Smithsonian Institute, writes us that he finds no mention of such "cannabalistic" tendencies in this species. They decreased steadily in numbers until the 27th of June, when they disappeared. In the latter part of September, however, two specimens believed to be of this species were taken, but we found no others, although the pool was carefully dredged. In May of the present year (1896), the pools being again filled, Apus lucasanus was taken again in the same places. Some specimens not yet identified, but probably of this species, were secured near Hudson, Colo., in the latter part of August. Three specimens of a species of Apus somewhat larger than lucasanus have also been taken, one of them in September, 1895, and the other two in June, 1896.

   In September, 1895, we found this species in several pools which were scattered for some distance along the draw mentioned above. So numerous were they that every cow track along the edges of the pools yielded eight or ten specimens. Two pairs were found in copulation. Specimens apparently of this species were taken on May 23 of this year in the same draw. These were probably young forms, for at a subsequent visit they were found to have increased in size. These specimens taken this year were of a bright red color, but faded badly when placed in alcohol. If individuals of this species are touched when swimming they immediately close their shells and drop to the bottom.

   In June, 1895, when Apus was first observed, some of this species were also seen, but none taken. Upon a subsequent visit they were found to have disappeared. In 1896 they occurred literally by millions in the pool north of De Witt, and quite a number were taken. Subsequently they were found in various grassy pools some distance north, but not a single one was taken in the draw previously mentioned. Egg sacs were observed in this and the above named species.

   One species of Branchinecta was also taken. These have no carapace and are quite different in appearance from the preceding. Out of the hundreds of Apus and large numbers of Eulimnadia and Estheria only five or six individuals of this variety were found, although diligently searched for. These were, in life, of a pale green color with carmine gonopoda, but fade quickly when placed in preservative.
   None of the species of Phyllopoda which occur in the west have been exhaustively studied, and those belonging to the Eulimnadia it is difficult to get identified with certainty. There is an opportunity, therefore, to find out many things about these short-lived and interesting creatures and discover facts pertaining to their life history, still obscure, which would be of great scientific interest.
   The writers will be pleased to receive any information concerning the occurrence of Phyllopoda in other parts of the state.

CONTINUED BIOLOGICAL OBSERVATIONS.

HENRY BALDWIN WARD.

   The wonderful advance given to scientific investigation by the work of the first naturalist who brought system and order into animal study was so great that students were long turned in the same direction and many of them were content to go no further. To most of them the mere discovery of some new animal was a matter of great importance, while its life, habits, and environment received little or no attention. The organism required simply a label before it should be laid away on the shelf of some museum as known. Nor was the mere study of anatomical detail much advance upon this standpoint. The information gained was isolated and unconnected with other facts that had been observed, and in the amassing of detail unity was lost sight of.
   Within the last few decades, however, there has been growing a desire to do more than to merely label a specimen or describe the details of its structure from some alcoholic material. It has come to have importance as a living thing, standing in close relations to other living things, influencing them and influenced by them; in other words, as a part of a whole which of itself must be studied.
   There are two ways in which the student may attack the problem of biological relations just suggested. He may investigate the sum of all the relations which pertain to a specific animal or those which are connected with a specific location with its sum of living things. The first problem is usually beyond the possibilities of the observer who does not possess considerable means for traveling or collecting through the medium of others, and the second, so far as it concerns a larger area, requires equally extensive collecting and an amount of literature which is not accessi-

ble to the majority of students. It is my desire here to call attention to a type of biological study which can be carried on in any locality and by any student with some hopes of being able to attain valuable results.
   Some years ago Forbes called attention to the fact that within a small lake we have a microcosm, a world dependent upon itself. Within this area is produced the entire amount of the food which is consumed by the animal life that inhabits the lake. The changes that take place are constant and yet constitute but a narrow circle. No area of land could be found of at all the same size, which would present equal possibilities for life, and at the same time so closely circumscribed that the problem would be confined to the area itself.
   The distribution of life within larger bodies of water has been the object of study to numerous investigators in the Old World, and in this country has been successfully prosecuted by Birge and Marsh in Wisconsin, Reighard in Michigan, Forbes in Illinois, and many others. Thanks to their researches we have learned much concerning the distribution of aquatic life from year to year, and from place to place. Into this object, however, it is not my purpose to go in detail. The information already gained will be of great value in attacking another aspect of the question. In the smaller areas of land and water the conditions are less variable and the problem in so far simpler. From the study of these limited environments, we must hope to attain to a better understanding of the biological laws which govern the change of material from the inorganic to the organic through its long series of steps. Every observer can find within easy reach a small pond which will serve as the object of his study. To it be must devote his undivided attention, and if he would succeed it must be mastered. The mere examination of the life it contains at the single time affords little information of value; hardly more useful are sporadic observations. The student must collect systematically and regularly throughout the entire year, keeping such record of conditions that he may be able to compare time with time. These collections must also be brought together in

such a way that they represent accurately the amount of life contained in a given amount of water under the observed conditions. From these data the student may determine the total quantity of living matter in the water at that time, and the relative amount of each different species. As the observations are extended he will be able to trace the rise and fall of a particular species, noting its first appearance and tracing it to its final disappearance. As thus gradually he records the history of the life in this microcosm it is evident that, continued long enough and carefully enough, he is recording the conditions which modify, which control the life itself.
   Evidently, then, from what has been said such studies have need of special apparatus, which must be at once permanent, portable, and precise. Hitherto in collecting material the investigator has made use of nets drawn vertically, horizontally, or obliquely through the water. They are, however, far from fulfilling any of the conditions satisfactorily, which have been set by investigators for such work. It was some years ago that in connection with more extended biological investigations on the Great Lakes the idea of a pump as a means of obtaining, from a specific point, an accurate quantity of water together with the life it contained, was first suggested to my mind and discussed with others. Since then the same idea has been carried into execution by others and the results obtained have been satisfactory. But of the apparatus thus far devised, it may be fairly said that its excessive weight and considerable cost renders it rather inaccessible to the ordinary investigator.
   In view of this fact, when suggesting to one of my more advanced students a topic along this line for investigation, I outlined to him a plan for a smaller pump which would be at once inexpensive and easily portable and which I hoped would give results satisfactory in precision as well. The plan which was submitted to him was carried out with some modification of detail and has proved its value in actual work, as he will explain to you in the next paper.

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