An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

tegrating complicated forms: it is the desire to aid the student
in forming these notions early, together with the writer's need
of a suitable exercise book for use in his classes, that lhas been
the reason for writing this paper.
The following is suggested as a course in fundamental principles and exercises: Elementary algebra, including progressions; convergence and divergence of series including the
elementary test theorems; sums of squares and cubes of
first n integers; undetermined coefficients and decomposition
of fractions; exponential and logarithmic series and logarithms;
elements of trigonometry; summation of series as in Chapter I,
this book; derived functions; theory of equations, graphs and
elementary limit theorems as stated in Arts. 7, 8, 9 and Theorem I,
Art. I4, MIcMahon and Snyder's Differential Calcilw's; permutations, combinations, binomial theorem; determinants, system
of linear equations, elimination, Sylvester's Method, discriminants; analytic geometry of plane and space; Chapter II, this
book; differential calculus proper, that is, the rules and formulas
of differentiation; Chapters III and IV; integral calculus followed by more complete courses.
For a simple demonstration of the logarithmnic series the following is suggested: e lo (i —x))- (-T —X)y; expanding by the exponential and binomial theorems and equating the coefficients of
y in the two expansions we obtain the logarithmic series. Of
course the limitations of this proof should be noticed. The
formulas for the sums of squares and cubes may be prosed by
induction. An early introduction to the factor and remainder
theorems with their application in drawing graphs and locating
roots of rational functions is advocated.
The kindness of Mr. H. H. Dalaker in reading proofs and
verifying examples is acknowledged.

UNIVERSITY OF MIINNESOTA
Mlarch, 1902