... in 1878, when Sylvester was hard at work on his main contribution to the inaugural volume of the 'American Journal of Mathematics', he undoubtedly exposed Franklin and his other listeners to his evolving ideas on the interrelations between invariant theory in algebra and the atomic theory in chemistry by punctuating his lectures, as usual, with open questions and unverified conjectures.

The expressions for the bilinear coordinates of a point in terms of its trilinear coordinates contain a common denominator which is a linear function of the latter. Whenever, therefore, the trilinear coordinates of a point are such as to make this function equal to zero, its bilinear coordinates are infinite; nor are they infinite under any other supposition: and hence the equation formed by putting this common denominator equal to zero is called the equation of the infinitely distant straight line. When we examine the corresponding expressions for the bilinear coordinates of a line, we do not arrive at any corresponding geometrical idea. These expressions, too, have a common denominator of the first degree; but the equation obtained by putting this denominator equal to zero represents simply the origin of coordinates, a point of no geometrical importance. I propose to construct a system of coordinates. in which the infinitely distant point shall hold a position similar to that held by the infinitely distant straight line in the bilinear system.

... find the number of different compounds that can be formed by $n m$-valent atoms and $(m - 2)(n + 2)$ univalent atoms; the word 'compound' being understood to mean any arrangement, whether continuous or not, in which every atom appears with exactly the number of bonds to which its valence entitles it; it being understood, moreover, that no two univalent atoms are connected with each other.

... in how many ways can $n - 1$ bonds be distributed among $n$ atoms, no atom having more than $m$ bonds attached to it?

The object of this paper is to give an account of the methods, due to Professors Cayley and Sylvester, of calculating the generating functions pertaining to binary quantics and thence determining the number of fundamental invariants and covariants of any order and degree. As it will not very greatly increase the length of the paper, I shall endeavour, besides giving the processes of calculation, to present a connected view, though not a complete discussion, of the subject.

Although he now served officially as his teacher's colleague, Franklin continued to participate in Sylvester's courses. In particular, he sat in on the number theory course which Sylvester had begun teaching in the fall of 1879 to a class including Arthur Hathaway, Christine Ladd, and Oscar Mitchell.

It will certainly not be unpleasant for you to hear that I was not the only person to be very interested in Mr Franklin's very original and ingenious proof. ... Mr Halphen, one of our most eminent young mathematicians, who has just won the Academy's Grand Prix in mathematics, found Franklin's method so remarkable that he lectured on it in one of the most recent sessions of the Philomathic Society. Please tell Mr Franklin that his talent is appreciated, as it deserves to be, by the mathematicians of the old world.

Professor Franklin was a very unusual and inspiring teacher. After a lapse of almost half a century former students still remember how clear and satisfying his lectures were. He was never content to expound a theory as it was developed by its author, and his hearers had the distinct impression that the matter being presented had been thoroughly digested by the lecturer and carried the stamp of his individuality and artistic nature. He was most conscientious and painstaking with his students, frequently going to great pains to answer a question whose import the student barely realised. His simple demeanour and dignity of person commanded the instant respect of his students, a respect which was never lost.

In his mathematics itself were reflected the wide interests of his inquiring mind. This was a definite fault of his genius. To leave a mark in mathematics requires an 'esprit de suite' which doggedly pursues to the bitter end some path into the unknown. The professional mathematician, on noting a characteristic flash of genius or originality in one of Franklin's short papers, may well wish that he had not been so versatile nor so interested in politics or world affairs; but such wishes are idle. In the short period of some fifteen years before he definitely abandoned mathematics for journalism, he published some thirty papers, most of which appeared in the 'American Journal of Mathematics' or the Johns Hopkins University circulars.

When President Gilman did me the honour to ask me to address you on this occasion, I felt some reluctance in undertaking the role of a Johns Hopkins Commencement orator. But upon considering that this is not only the first, but in all probability the last occasion on which I am to appear before the students and friends of Johns Hopkins University, I felt that I could not let it go by. As most of you probably know, I am about to make a change in my occupation - in one aspect perhaps the most extreme change that it is possible for a man to make in his mental atmosphere. Mathematics is the domain of the most exact and rigorous thinking of which the human mind is capable; I fear you will agree only too readily with me in pronouncing journalism to be the field in which loose and inexact thinking is most at home. This is in a great measure unavoidable from the nature of the case; and yet the contrast I have just mentioned seems to give appropriateness to the subject to which I shall venture to ask your attention for a few minutes - the need of exact thinking in the discussions of actual life.