Miss Sanderson was gentle-mannered, of brilliant intellect, an exact student, broad-minded, self-reliant, and courageous.

Of this original and valuable thesis a very brief extract was printed in the 'Annals of Mathematics', Series 2, Volume 13, 1911, pages 36-39. This work might well have served for her doctor's thesis; but she was quite willing to undertake a new investigation in a wholly different field.

The term function is here used to denote a rational integral function of y with integral coefficients. Employing a fixed integer m and a fixed function P(y), we shall say that two functions are congruent modulis m and P(y) if their difference can be given the form mq(y) + P(y)Q(y); also that f(y) has an inverse $f_{1}(y)$ if $f(y).f_{1}(y)$ is congruent to unity modulo m, P(y). Then f(y) and f(y) + k(y)P(y) have the same inverse, so that we may restrict attention to functions of degree less than the degree r of P(y). We proceed to prove the
Theorem.
If P(y) is of degree r and is irreducible with respect to each prime factor of m, a function R(y) of degree < r has an inverse modulis m and P(y) if and only if the greatest common divisor d of the coefficients of R(y) is prime to m.

... was entitled "Formal Modular Invariants with Applications to Binary Modular Covariants," and appeared in the 'Transactions' of the American Mathematical Society, Volume 14, 1913, pages 489-500. This paper is a highly important contribution to this new field of work; its importance lies partly in the fact that it establishes a correspondence between modular and formal invariants. Her main theorem has already been frequently quoted on account of its fundamental character. Her proof is a remarkable piece of mathematics.

Miss Sanderson's single contribution (1913) to modular invariants has been rated by competent judges as one of the classics of the subject.

We see clearly just how the difference in the definitions of formal and modular invariants affects the actual computations. Dickson has given a very simple and elegant theory of modular invariants. No theory has been developed for formal invariants. However, there exists between the two subjects an interesting and important relation, which I shall develop in what follows. I take this opportunity to express my gratitude to Professor Dickson for his interest and many helpful suggestions, in particular for the present formulation of this introductory section.

MILDRED SANDERSON was born in 1889 in Waltham, Massachusetts, where she received her secondary school education. She took the degree of Bachelor of Arts at Mount Holyoke College in 1910, the degrees of Master of Science (1911) and Doctor of Philosophy (1913) at the University of Chicago, doing her major work in mathematics, and her minor work in astronomy.

Sanderson was an instructor at the University of Wisconsin during the first semester 1913-14 but left in February 1914 when she became ill with pulmonary tuberculosis. She died at age twenty-five in a hospital in East Bridgewater, Massachusetts, the following October. Services were conducted in her family's home in Waltham by a pastor of the First Baptist Church, and she was buried in Mt Feake cemetery in Waltham.

The remarkable mathematical ability and originality shown by Miss Sanderson in her master's and doctor's theses and the very unusual ease with which she assimilated ideas in all branches of pure and applied mathematics, combined with her enthusiasm for that science, gave full promise of a highly successful career for her in research. Her death on October 15, 1914, only a year after completing her graduate studies, was not only a distinct loss to progress in mathematical research in America, but was a very keen blow to her fellow students, to all of whom she had endeared herself by her most lovable personality. ... If I may be permitted to add my personal tribute to the universally expressed tribute to her remarkable ability, it would be to say that she was my most gifted pupil.