# Book:L. Harwood Clarke/A Note Book in Pure Mathematics

## L. Harwood Clarke: A Note Book in Pure Mathematics

Published $\text {1953}$, William Heinemann Ltd.

### Contents

Foreword
$\text {I}$. ALGEBRA
Permutations and Combinations
The value of ${}_n P_r$
The number of ways of arranging $n$ things in a line
The number of ways of seating $n$ people at a circular table
The number of ways of threading $n$ beads on a wire
The value of ${}_n C_r$
The number of ways of arranging $n$ things in line if $p$ are alike of one kind and $q$ are alike of another kind
The number of ways of dividing $\paren {p + q + r}$ things into three unequal groups, the first to contain $p$ things, the second $q$ things and the third $r$ things
The number of ways of dividing $3 p$ things into three equal groups each containing $p$ things
The number of selections from $n$ things if any number may be taken
The number of selections from $n$ different things, $p$ similar things of one kind and $q$ similar things of another kind, if any number may be taken
${}^n C_r = {}_n C_{n - r}$
${}^n C_r + {}_n C_{r + 1} = {}_{n + 1} C_{r + 1}$
The Method of Induction
The Binomial Theorem
Relations between coefficients
Negative and fractional indices
Approximations
The greatest term
Partial Fractions
Summation of Series
The sum of the squares of the first $n$ integers
The sum of the cubes of the first $n$ integers
Power series
The Exponential and Logarithmic Series
Applications to summing series
The Remainder Theorem
Roots of Equations
The condition that $\paren {a x^2 + b x + c}$ should be positive for all values of $x$
The condition for a common root
The condition for a repeated root
Relations between roots
To form the equation whose roots are symmetrical functions of each of the roots of a given equation
To form the equation whose roots are any symmetrical functions of the roots of a given equation
The sum of the powers of the roots of a given equation
Finding numerical roots
The function $\dfrac {a x^2 + b x + c} {A x^2 + B x + C}$ when $x$ is real

$\text {II}$. CALCULUS
Differentiation
Differential coefficient
Standard differential coefficients
Double function
Product
Quotient
Variable index
Inverse ratios
Maximum, Minimum and Point of Inflection
Velocity and Acceleration
Tangent and Normal
Parametric Equations
Differentiation $n$ Times
Leibnitz' Theorem
Maclaurin's Theorem
Taylor's Theorem
Integration
Algebraic integration
Powers of cos and sine
Useful substitutions
Integration by parts
Definite Integrals
Integral of $1 / x$
Area, Volume and Centre of Gravity
Arcs
Polar Coordinates
Moments of Inertia
Reduction Formulae
Hyperbolic Functions
Differential Equations

$\text {III}$. ANALYTICAL GEOMETRY
The Straight Line
Point dividing $AB$ in a given ratio
Centre of gravity of a triangle
The area of a triangle
Forms for the equation of a straight line
The angle between two lines
The length of the perpendicular
The equations of the angle bisectors
Lines through the intersection of two given lines
Pairs of Straight Lines
Their equation
The angle between the pair
The equation of the angle bisectors
The condition for a pair
The Circle
Its equation
The tangent of gradient $m$
The tangent at $\tuple {x', y'}$
The polar of $\tuple {x', y'}$
Orthogonal circles
The length of a tangent
The Parabola
Its equation
The tangent of gradient $m$
The normal
The equation of a chord
The tangent at $\tuple {x', y'}$
The locus of the foot of the perpendicular from the focus to a tangent
The locus of the intersection of perpendicular tangents
The polar of $\tuple {x', y'}$
The feet of the normals from a point
The locus of the mid points of parallel chords
The chord of mid point $\tuple {x', y'}$
The Ellipse and Hyperbola
Their equations
The tangent of gradient $m$
The auxiliary circle
The director circle
The tangent at $\tuple {x', y'}$
The normal at $\tuple {x', y'}$
The polar of $\tuple {x', y'}$
The pole of $l x + m y + n = 0$
The locus of the mid points of parallel chords
Conjugate diameters
The chord of mid point $\tuple {x', y'}$
The Ellipse
The eccentric angle
The line joining $\alpha$ and $\beta$
Eccentric angles at the ends of conjugate diameters
The Hyperbola
The asymptotes
Parametric representation
The Rectangular Hyperbola
Its equation
The tangent at $\tuple {x', y'}$
The polar of $\tuple {x', y'}$
The chord joining $m$ and $l$
The locus of the mid points of parallel chords
The chord of mid point $\tuple {x', y'}$
The equation of a hyperbola with given asymptotes
The General Conic
The conditions for: a pair of straight lines, a circle, a parabola, an ellipse, a hyperbola, a rectangular hyperbola
A conic through the intersections of two given conics
The polar of $\tuple {x', y'}$

$\text {IV}$. PURE GEOMETRY
Plane Geometry
The incentre
The circumcentre
The orthcentre
The centre of gravity
The centre of similitude
Apollonius' Circle
Ptolemy's Theorem
The Euler Line
The Nine-Point Circle
Ceva's Theorem
Menelaus' Theorem
The Simson Line
The radical axis and coaxal circles
Solid Geometry
The plane
Skew lines
Generators
The angle between a line and a plane
The angle between two planes
If a line is perpendicular to each of two intersecting lines, it is perpendicular to their plane
The tetrahedron
Its circumscribing parallelepiped
Its circumscribing sphere
The common perpendicular to two skew lines
The intersection of a sphere and a plane
The intersection of two spheres
Orthogonal Projection
Lines
Centre of Gravity
Areas
The circle
Geometrical properties of the ellipse

$\text {V}$. TRIGONOMETRY
Definitions of the Ratios
Special Angles
Complementary Angles
Angles Larger than $90 \degrees$
The Cosine Formula and the Sine Formula
Tangents of Sum and Difference
The Product Formulae
Factorization of Sums of Sines and Cosines
The Auxiliary Angle
Solution of Equations
Solution of Triangles
Inverse Ratios
Half Angle Formulae
Area of the Triangle
The Median and the Centre of Gravity
The Orthocentre
The Angle Bisector
The Pedal Triangle
The Circumcircle
The Incircle
The Ex-circles

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