Professor Dowling's instruction was characterised by an exceptional clarity of exposition which, with his magnetism, geniality of manner, and interest in his students, made him a universally popular and beloved teacher. He was a man of unusual breadth of interests: a talented musician, an eager and well-informed botanist, and a student of the humanities. His knowledge of the language and literature of Italy, where he studied under the geometer Segre, was exceptional; and he was an ardent reader and lover of poetry. These talents explain, in part, the bonds of sympathy that united him with so many colleagues and students. But above all his attraction was due to his personal qualities, his cheerful disposition and his kindly character. His influence in these private associations was no less great than in his admirable public service as a teacher, and he will long be remembered and missed by his friends.

A party of tourists left Madison in June 1905 for a summer's trip in Europe, under the guidance of Professor L W Dowling, assistant professor of mathematics. The other members of the party are Mrs Dowling, Mrs Eliza Allen and daughter Miss Florence E Allen, (1900), instructor in mathematics; Miss Merle S Pickford (1902), instructor in history in the Eau Claire High School; and Miss Jennie Sherrill (1902), of Belvidere, Ill., instructor in history in Belvidere high school. The party travels together through England, Holland and Germany to Switzerland, where Mrs Allen and daughter will return to Bonn, Germany, to meet Mr C E Allen, (1899), Mrs Allen's son, who, with his wife, has been in that city attending the University for the past year, engaged in the study of botany, of which he is instructor at the University. The other members of the party will proceed to Italy, where Dr and Mrs Dowling expect to remain until 1 June 1906. Dr Dowling will spend a year in study at Turin, Italy. Miss Pickford and Miss Sherrill will sail for home by way of Naples and the Mediterranean route, 26 August, on the steamship Canopic, White Star Line, landing at Boston. Mrs Allen and daughter and Mr Allen and wife will sail from Hamburg 1 September reaching here before the opening of the University next fall.

Posts have been received from Miss Merle Pickford who with Dr and Mrs Dowling, Mrs Eliza Allen and daughter, Miss Florence Allen arrived safely at Liverpool after a pleasant voyage, and who presumably are now in London.

In an article entitled "Razionalita della Involuzioni Piane," Castelnuovo proved the rationality of all involutions in the plane. In his notation, a plane involution is defined by four equations between $(\alpha, \beta, \gamma)$, the points of a plane, and $(x_{1}, x_{2}, x_{3}, x_{4})$, the points on a surface in space of three dimensions:

$\rho x_n = f_n(\alpha, \beta, \gamma) (n = 1, 2, 3, 4)$,

the functions $f_{n}$ being rational forms of the same degree. To a given point in the plane corresponds uniquely a point of the surface; but to a given point on the surface correspond, in general, not one point, but several points, in the plane. When to a point on the surface corresponds a group of two or more points in the plane, these groups are said to constitute an involution in the plane. From the point of view of the plane itself, an involution means the establishment of an $∞^{2}$ series of point groups, each point of the plane belonging to one and only one group, and each point determining the remainder of the points associated with it to form a group. The important theorem that all such involutions are rational signifies that a one-on-one correspondence may be established between their groups and the points of a plane. Hence, if the points in the above equations range over a surface I, each point of I giving rise to a group of $n$ points in the plane $\pi$, it is always possible to establish a mutually unique correspondence between the points of I and the points of a plane $\pi'$. Thus every involution may originate in a plane π' so that to each point of $\pi'$ corresponds exactly one group of the involution in π. The problem may then be transferred from the study of surfaces to the study of systems of curves in the plane, as follows.

On account of the rationality of the surface, the above equations may be replaced by relations connecting the points $(\xi, \eta, \zeta)$ of $\pi'$ with the points $(x,y,z)$ of $\pi$:

ρ ξ = U(x,y,z),
ρ η = V(x,y,z),
ρ ζ = W(x,y,z),

for ξ:η:ζ = U:V:W.

To a point $(\xi_{1}, \eta_{1}, \zeta_{1})$ corresponds in $\pi$ a group of points determined as the variable intersections of $U/ \xi_{1} = V/ \eta_{1} = W/ \zeta_{1}$, i.e., the intersections outside of the fundamental points common to $U, V$, and $W$.

Thus every involution may be generated by a linear net of curves

$\lambda U + \mu V + \nu W = 0$.

The involution will be characterised as an $I_n$, $n$ being the number of points in a group, or the number of intersections of two curves of the net outside of the base points; $n$ will be call the "order" of the involution.

There is thus established a general theory of involutions of any order and any kind. In this theory, as formulated by Castelnuovo, reference is made to the so-called "cyclic" involutions, i.e., those in which the transformation is effected by a periodic birational transformation, either a collineation or a periodic Cremona transformation. The theory of the cyclic periodic transformations of any order has been developed by Kantor and Wiman. The involutions of order 2 were first classified by Bertini, these, as is known, being entirely cyclic. There has apparently as yet been no attempt to distinguish between the geometric construction of the cyclic and the non-cyclic types, a distinction which would rise for the first time in the case of the $I_3$. It is the purpose of this paper to discuss the construction of an $I_3$ from the point of view of nets of curves, and to differentiate, in the case of nets of deficiency 0, 1, and 2, between those which lead to cyclic and to non-cyclic involutions. for the systems of curves of the above deficiencies the cyclic cases will be found, and in each case the analytic expression of the periodic transformation set up.

G N Bauer and N L Slobin have investigated curves of the type

(1) $\Phi (x, y) · e^{\Psi(x,y)} + X(x, y) = 0$

where $\Phi, \Psi$ and $X$ are entire rational functions with algebraic coefficients. They have found that at least one coordinate of every point of this curve is transcendental, except for certain trivial points that are obtained by setting $\Phi$ and $X$ or$\Psi$ to zero. The author now discusses this curve in more detail and finds, for example, the theorem that the curve has no transcendental multiple point. They also investigate the case where $\Phi, \Psi$ and $X$ are functions of the same function $\theta(x, y)$, where

(1) degenerates into a family of curves

$\theta(x, y) = c$,

where $c$ can only take on values ​​of a special nature.

In a series of recent papers, [University of Washington professor] R M Winger (1918, 1919 and 1920) has developed various features of the satellite theory of the plane cubic by use of the parameter form of representation and involutions on rational curves. Classic theorems and extensions of satellite properties are proved with great ease and beauty.

A prominent feature of these papers is the discussion of systems of contact curves. Of special interest is Winger's conic $N$, the envelope of lines joining pairs of contact of tangents from points of the rational cubic, and tangent to the cubic at the sextactic points. This conic is one of a pencil of conics which touch the nodal tangents where they meet the line of inflexions. Since both the cubic and each conic of this pencil are invariant under a dihedral $G_{6}$, the pencil and the conic $N$ play an important part in the construction of special and general sets of the group.

The problem of closure of the tangential process on an elliptic cubic or higher curve has been extensively discussed from the number-theory standpoint, (for example by Sylvester, Story, Picquet, and Porter) leading to the determination of sets of real, rational points on a curve. With this has been developed the configurations of polygons, both inscribed and circumscribed to a curve of third order or higher.

The present paper takes up the problem of determining those points on a rational cubic which coincide with their nth tangentials from the more general projective standpoint. It is found that the pencil of conics above described is of fundamental importance.

On the rational plane curve of third order, those points are successively sought that coincide with its first, second, third, ... tangent points. The geometric significance of such point groups for the curve is then discussed.

The publication of such texts as this may be taken as a hopeful sign - not that the curriculum is becoming over commercialised, but that the commercial world is finding it increasingly necessary that its people shall be trained in logical mathematical thought.

Of course there will always be some women who should go in for a Ph.D. - some because it will be an actual necessity to qualify them for one of the occasional - very occasional - openings in college and university positions, some because of the leisure they may have to follow a congenial pursuit. But on the whole I see no great encouragement to be had from past experiences and observation. I do not believe that there is or will be a great future for any but a few in this field. At present, it seems to me, as I look about this campus, that in all strictly academic fields (not those special to women) that there is a decided drop in the number of women engaged. That may be peculiar to this economic phase, but I look for it to continue for some time to come.