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

Contents

 * Foreword


 * $\text {I}$. ALGEBRA


 * 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
 * ${}^n C_r = {}_n C_{n - r}$
 * ${}^n C_r + {}_n C_{r + 1} = {}_{n + 1} C_{r + 1}$
 * ${}^n C_r + {}_n C_{r + 1} = {}_{n + 1} C_{r + 1}$




 * Relations between coefficients
 * Negative and fractional indices
 * Approximations
 * The greatest term
 * The greatest term




 * The sum of the squares of the first $n$ integers
 * The sum of the cubes of the first $n$ integers
 * Power series
 * Power series


 * Applications to summing series
 * Applications to summing series




 * The quadratic function
 * 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
 * The function $\dfrac {a x^2 + b x + c} {A x^2 + B x + C}$ when $x$ is real


 * $\text {II}$. CALCULUS


 * Differential coefficient
 * Standard differential coefficients
 * Double function
 * Product
 * Quotient
 * Variable index
 * Inverse ratios
 * Inverse ratios










 * $n$








 * Algebraic integration
 * Powers of cos and sine
 * Useful substitutions
 * Integration by parts
 * Integration by parts




 * $1 / x$
















 * $\text {III}$. ANALYTICAL GEOMETRY


 * 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
 * Lines through the intersection of two given lines


 * Their equation
 * The angle between the pair
 * The equation of the angle bisectors
 * The condition for a pair
 * The condition for a pair


 * 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
 * Coaxal circles and radical axes
 * Coaxal circles and radical axes


 * 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 chord of mid point $\tuple {x', y'}$


 * 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 chord of mid point $\tuple {x', y'}$


 * The eccentric angle
 * The line joining $\alpha$ and $\beta$
 * Eccentric angles at the ends of conjugate diameters
 * Eccentric angles at the ends of conjugate diameters


 * The asymptotes
 * Parametric representation
 * Parametric representation


 * 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 equation of a hyperbola with given asymptotes


 * 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 $\tuple {x', y'}$
 * The polar of $\tuple {x', y'}$


 * $\text {IV}$. PURE 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
 * The radical axis and coaxal circles


 * 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
 * The intersection of two spheres


 * Lines
 * Centre of Gravity
 * Areas
 * The circle
 * Geometrical properties of the ellipse
 * Geometrical properties of the ellipse


 * $\text {V}$. TRIGONOMETRY








 * $90 \degrees$






































 * ANSWERS



Source work progress

 * From :


 * : $\text V$. Trigonometry: Exercises $\text {XXXII}$: $13$.


 * From :


 * : $\text {III}$. Analytical Geometry: The Straight Line: The angle between two lines: Example $\text{(ii)}$


 * From :


 * : $\text {II}$. Calculus: Integration: Useful substitutions


 * : $\text I$. Algebra: The Binomial Theorem: Exercises $\text {III}$: $1 \ \text {(d)}$