Date of Award

January 1940

Degree Type



The faint beginnings of the theory of symmetric functions can be traced as far back as the sixteenth century. The starting point seems to be with the eccentric Italian mathematician of the middle sixteenth century, Girolamo Cardano (1501--1576) whose name has been associated with the first solution of the cubic equation [1;361]. From him flows a stream of eminent mathematicians, Euler, Wallis, Newton, Hirsch, Girard, Waring, Hammond, and others who, like Cardan, with the theory of the roots of algebraic equations in one or more variables as an incentive did much in the development of the theory of symmetric functions. In this paper the author can only praise these renowned and worthy scholars of previous generations who have provided such a noble heritage for students of to-day. However, this treatise is concerned with the theories of more recent writers on the theory of symmetric function---such writers a MacMahon, Carver, O'Toole, and Dwyer, who introduced improvements of notation and symbolism, the lack of which had stunted the growth of the theory of symmetric functions since its birth in the sixteenth century. This ''algebraic shorthand'' is very essential in the development of formulas and tables that are of the greatest value to the statistician and of the deepest interest to one not necessarily concerned with statistics.