The elementary properties of generators of a ruled surface, and the existence of a line of striction when the surface is skew, are well known to readers of this journal. We propose to show that many of these properties do not belong exclusively to ruled surfaces; but that a family of curves on any surface possesses a line of striction and a focal curve or envelope, though these are not always real. When the surface is developable, and the “curves” are the generators of one system, the focal curve is the edge of regression. We shall also see that the properties to be established for a family of curves on a surface are analogous to some of the leading properties of congruences of curves in space, the line of striction corresponding to the surface of striction or orthocentric surface, and the focal curve to the focal surface of the latter.

According to the standard-text-books, an asymptote is a line which meets a curve at two points at infinity, but is not wholly at infinity. Regarded as a flash of mathematical wit the definition is clever and illuminating, but it is far too subtle for a raw student. He may have learnt that to call parallel lines “lines that meet in a point at infinity” is merely a Pickwickian way of saying that they do not meet at all, but, being in a plane, had (so to speak) their chance of meeting and just failed to take it; but even if he understands that, he may still be mystified here. For the asymptote and the curve meet, he is told, at two points at infinity, and who could guess what complicated tragedy of failure that euphemism conceals?

IT is well known that, while the shortest course between two points on a sphere is a great circle, the course which is easiest for navigation is a rhumb line, i.e. a course which cuts all the successive meridians at the same angle, so that the ship always steers in a direction making an angle of the same number of degrees with due north or due south. Such a course is represented by a straight line on a Mercator’s chart, which thus makes it appear more direct than it really is. On a great circle, though the course is the shortest, you are (except on the equator or a meridian) continually altering your direction as measured by the angle which it makes with the north. Between any two points there is thus a difference in favour of great-circle sailing; but to demonstrate rigorously the greatest value which this difference can possibly take is a somewhat difficult and attractive mathematical problem. I have heard it stated that it is not susceptible of rigorous mathematical treatment. The following is suggested as giving a complete solution of the question: it would be interesting to learn whether any shorter or more direct method is available, which makes no unwarranted assumption. For the abstract treatment of the problem (though indeed in the Pacific or the Southern Ocean fairly extended courses are practically available) we suppose the globe cleared of all such inconvenient encumbrances as continents or islands: moreover, we treat it as a perfect sphere; and to word the question more exactly we should say: Find two points such that the difference between the shortest great-circle course and the shortest rhumb-line course joining them is a maximum; for the great circle through two points will (in general) consist of a major and a minor arc, and on the other hand a rhumb line might start the longer way round the earth, or indeed make any number of revolutions about the pole before reaching its objective. In either case it is the shortest course of the kind with which we deal.

1. The English student of Mathematics finds in his text-books a proof that the uniform convergence of a series of continuous functions is a sufficient condition for term-by-term integration. He knows that there are convergent series which cannot be integrated term-by-term,f and that there are series, not uniformly convergent in the interval of integration, which can be thus integrated. The last remark applies particularly to the Fourier’s Series of an ordinary function with discontinuities.

I have been asked to write a short account of the present-day work in the algebra of invariants for the benefit of those who have not made a special study of the subject. A good summary account can be found in the various continental works of reference, and particularly in the Encyklopädie der Mathematischen Wissenschaften, Band III. Teil iv. 7, written by Dr. R. Weitzenböck, and brought up to about tho date 1920 with tolerable completeness, both in subject-matter and in references. Nothing has, however, been published in the last thirty years quite so good and readable as Meyer’s Bericht, which reviewed the first fifty years of progress from the time when Boole inaugurated the theory in the Cambridge Mathematical Journal, iii. pp. 1-20, of the year 1841. As things have turned out, and especially because of the impossibility of buying separate parts of the Encyklopädie without embarking on the whole section (Geometry in this particular case), it seems a pity that we have no machinery in England for the systematic issue of corresponding reviews, perhaps in the form of this Bericht, to cover by degrees the field of present-day mathematics.

The next International Congress is to be held at Bologna in September of next year. For the first time since the Cambridge Congress of 1912, all differences of nationality are to be disregarded, and mathematicians who believe that in spite of all that has happened since 1912 international cooperation is possible, or even that because of all that has happened international cooperation is imperative, will be glad to have ample warning in order not to make plans that might inadvertently interfere with their attendance.

The title of this paper was chosen before I had put my thoughts together, and even before I had considered thoroughly whether it were possible to make any general conclusions at all. I hope I shall not be accused afterwards of making something out of nothing, or of seeing connections where none exist. Perhaps it will be that, instead of having a vague notion that mathematics and morals are strangers or opposites, we may arrive at the end of the evening with a reasoned conviction that they are indeed so. But that the subject is not unprofitable to discuss I am strengthened in believing by noticing that Henri Poincaré, perhaps the greatest mathematician of the last fifty years, in the closing years of his life wrote an essay on the “Relations of Morals to Science,” which is included in the volume of his last thoughts.

In 1679 Leibniz writing to Huyghens, said : “I am not satisfied with the Algebra (of coordinates), inasmuch as they give neither the shortest ways nor the most beautiful theorems of Geometry; and think that there ought to be a more direct way of dealing with the elements of a figure.” He gave a sketch of his ideas, which, however, formed no real contribution.

I have been asked to speak about developments in Mathematical Logic since the publication of Principia Mathematica, and I think it would be most interesting if, instead of describing various definite improvements of detail, I were to discuss in outline the work which has been done on entirely different lines, and claims to supersede altogether the position taken up by Whitehead and Russell, as to the nature of mathematics and its logical foundations.

Considering the setting for this year’s meeting of the British Association, to doubt whether Section A will provide a subsectional programme attractive to readers of the Gazette is to overlook the notorious energy of Yorkshire mathematicians. Details are not yet available, but people who enjoy social encounters and holiday mathematics can be sure of a pleasant week at Leeds from August 31st to September 7th.