136 ROYAL SOCIETY OF CANADA 
cartes’ Theorem.—Complex numbers. De Moivre’s Theorem. Trig- 
nometric resolution of the binomial equation. 
Functions: Function of a real variable, graphic representation, 
continuity.—Definition and continuity of the exponential function and 
of the logarithmie function. Limit of (1 + +)* when m increases 
indefinitely in absolute value.—Derivative of a function; slope of the 
curve represented. Derivative of a sum, of a product, of a 
quotient, of an integral power; of a function of a function. Derivative 
of a* and of log x. — Use of logarithm tables and of the slide rule. 
—Rolle’s Theorem, law of finite increments, graphic representation.— 
Functions of several independent variables, partial derivatives. Law of 
finite increments. Derivative of a compound function. Derivative 
of an implicit function (admitting the existence of this derivative). 
—Employment of the derivative for the study of the variation of a 
function; maxima and minima. Primitive functions of a given function, 
their representation by the area of a curve. 
Functions defined by a power series with real coefficients. Interval of 
Convergence: Addition and multiplication. In the interior of the 
interval of convergence one obtains the derivative or the primitive 
functions of the function, on taking the series of derivatives or of the 
primitive functions (functions which pass to the extremities of the 
interval are not considered).—Examples:—developments in series of 
ce ins, arctan x, log (1—x), log 5%. Exponential series. Binomial 
series. The equations y'— y, and y'(1 + x) — my serve to determine 
the sum of two series.—Development into series of a*, of aresin x. 

Curves whose equation is soluble or insoluble with regard to one of the 
co-ordinates: Tracing. Equation of the tangent at a point; sub-tangent. 
Normal, sub-normal. Concavity, convexity, points of inflexion. 
Asymptotes. Application to simple examples and in particular to the 
conics and to those curves of which the equation is of the second degree 
with respect to one of its co-ordinates. 
Curves defined by the expression of the co-ordinates of one of their 
points as function of a prameter: Tracing. Numerical examples. 
The curves of the second order and those of the third order with a double 
point are unicursal. 
Curves defined by an implicit equation: Equation of the tangent 
and of the normal at a point. Tangents at the origin in the case where 
the origin is a simple point or a double point. Discussion of the asym- 
ptotes in the case of numerical examples of curves of the second and of 
the third order. 
Curvature. Envelopes. Developables. 
Polar Co-ordinates: Their transformation into line co-ordinates. 
Equation of a right line.—Construction of curves; tangents, asymptotes 
