INDEX

[The numbers refer to the pages]

Absolute convergence, 189, 190.

Absolute value, 78, 145.

Addition theorems, 64, 105, 123, 193, 198, 199.

Algebraic factors, 219.

Ambiguous case, 2.

Amplitude, 145, see principal value.

Angle-bisectors in triangle, 11.

Angle measurement, 119.

Approximation, 82, 95.

Area, hyperbolic, 52.

of quadrilateral, 24, 27, 28.

Argand diagram, defined, 145.

z₁±z₂, 148.

z₁z₂, z₁ ÷ z₂, 151.

z², 1/z, 152, 153.

zⁿ, 165 to 167.

Argument, 145.

Base of logarithms, natural, 63.

generalised, 241, 253.

Binomial Series, 142 ii. and Ex. 3, 253.

Centroid, 10.

ch, sh, behaviour of, 109.

calculus applications, 107, 110.

formulae for, 105.

generalised, 198.

inverse, 110, 256.

Circular functions, see cos.

Circumcentre, 4.

cis θ, 152.

Complex number, amplitude, 145, 146.

conjugate, 143.

definitions, 137 to 139.

difference, 139, 148.

first and second parts, 143.

geometrical representation, see Argand.

i, 140, 141.

Complex number, manipulation, 141.

modulus, 145.

modulus-amplitude, 145.

nomenclature, 143.

notation, 140, 152.

product, quotient, 139, 151, 152.

sum, 138, 148.

Complex variable, functions of

geometrical representation, see Argand.

principal value, 146, 164, 165.

exp, 191.

log, 241, 253.

sh, ch, etc., 198.

sin, cos, etc., 197, 199.

sin^–1, sh^–1, etc., 256.

Compound Interest law, 93.

Convergence, 77.

absolute, 189, 190.

circle of, 247.

conditional, 190.

cos (A + B), 123.

cosh, see ch.

cos θ, sin θ, differential of, 79.

expansion of, 80, 81.

exponential form, 194.

factors of, 223.

generalised, 197.

cos nθ, in factors, 223.

in terms of c and s, 172.

in terms of c or s, 178.

cosⁿθ, cos^pθ sin^qθ, in multiple angles 169.

Cotes’ properties, 228.

Cubic equations, 44.

Cyclic quadrilateral, 24.

De Moivre, property of, 228.

theorem of, 151, 162 ;

and applications to

expansions, 169, 172.

factors, 219, 226.

De Moivre, powers and roots, 165.

solution of equations, 167.

summation of series, 174.

Distances between points in triangle, 15.

e, defined, 63.

evaluated, 91.

irrationality, 92.

series, 91.

E-centre, 4.

Elimination, 270.

Equations, approx. solution, 82, 95.

construction of, 204.

cubic, 44.

functions of roots of, 206, 208.

graphical solution, 38.

trigonometrical, 34.

Errors, 20.

Essentially distinct roots, 212.

Euler’s constant, 70.

Expansions, polynomials, etc.,

cos nθ, sin nθ, 172, 178.

cosⁿθ, sinⁿθ, 169.

tan nθ, tan ∑θ, 172, 173.

Expansions, power series

(1 + x)^–1, 77.

(1 + z)^–1 (1 + z)^m, 191, 253.

ch, sh, 104, 198.

cos, sin, 79, 198.

e^x, exp z, 90, 191.

log, 84, 85, 245.

tan, 82.

tan^–1, 88.

Exponential function,

defined, 64, 192.

differential of, 64.

expansion of, 90.

limit form, 93, 75.

Factors, algebraic, 219.

cos θ, sin θ, 223.

Cotes and de Moivre, 228.

fundamental theorem, 142, 219.

series and products, 228.

sin nθ, etc., 222, 227.

trigonometrical, 221.

xⁿ ± 1, 219, 220.

x²ⁿ – 2xⁿ cos nα + 1, 226.

Feuerbach’s theorem, 16.

Functions, circular, 79, 197.

exponential, 64, 192.

‘hyp,’ 52.

Functions, hyperbolic, 104, 198.

inverse, 46, 64, 88, 110, 256.

log, 63, 84, 241, 247, 253.

many-valued, 241.

Geometric progression, 77, 191.

Gregory’s expansion, 88.

Hyperbolic functions, 52, and see

ch.

i, 140, 141.

Identities, 263.

IH², I₁H², IN, I₁N, 16.

In-centre, 4.

Indices, zⁿ, 140.

z^p/q, 162.

z^w, e^iα, 252, 253.

Inequalities, cos, sin, 80.

exp, 71, 74, 97.

log, 67, 71, 73, 74.

miscellaneous, 57.

trigonometrical, 274.

Infinite Integrals, 53, 54, 57 (No. 17), 75, 76.

Infinite Products, 223, 240.

Infinity, Sum to, 77, 190.

Integration, 63, 64, 79.

Inverse Functions (see also Functions),

differentiation of, 157 (No. 16).

principal values, 155.

tan^–1m ± tan^–1m′, 47, 156 (No. 12).

Limits for n→∞,

xⁿ, xⁿ/n!, 78.

(1±x/n)ⁿ, 93.

(cos x/n)ⁿ, 70.

, 69.

– log n, 69.

Limits for, x→∞,

hyp x, 53.

(log x)/x^p, 68, 71 (No. 11), 73 (No. 25).

Limits for y→0,

(e^y – 1)/y, 69.

hyp | y |, 54.

{log(1 + y)}/y, 68.

y log | y |, 68.

Logarithms, base e, 63.

differentiation, 63.

inequalities, 67.

Logarithms, integration, 67 (No. 45).

log w, log_zζ, 241, 253.

Machin’s formula, 89.

Many-valued functions, 47, 155, 241.

Mass-centre theorem, 10.

Maxima and Minima, 274.

Median, 11.

Modulus, 78, 145, 148.

Modulus-amplitude form, 145.

Nine-point circle, 6, 16.

Number, 137, and see Complex.

OH², OI² OI₁², 15.

Ordered pairs, 138.

Orthocentre, 5.

Osborn’s rule, 105.

π, evaluation, 89.

product, 223.

π²/6, 209, 228.

Partial fractions, 231,

Pedal triangle, 5.

Polar circle, 6.

Power series, 77.

Powers, see Indices.

Powers of cos θ, sin θ, 169.

Principal values,

amplitude, 146, 156 (No. 8).

(cis θ)^p/q 164.

cos^–1x, etc., 155.

Log w, Log(1 + w), Log_zζ, 241, 247, 254.

range of, 155.

Tan^–1z, 258.

z^w, (1 + z)^m, 252, 254.

Projection, cos, sin(A + B), 123.

points and lines, 118.

summation, 125.

Ptolemy’s theorem, 25, 27,

Quadrilateral, circumscribable, 27.

cyclic, 24.

general, 27.

R, r, r₁, ρ, 4, 5, 6, 25.

Roots, essentially distinct, 212.

Roots of equations, 204.

Rutherford’s formula, 89.

Series (see also Expansions)

∑x^r, ∑z^r, 77, 191.

∑x^r/r!, ∑z^r/r!, 90, 191.

∑r^–2, 209, 228.

∑r^–4, 212 (No. 21).

∑ cos (α + rβ), ∑ sin (α + rβ), 125, 127.

∑x^r cos rθ, ∑x^r sin rθ, 174.

∑( – 1 )^n–1ρⁿn^–1 cos (or sin)nϕ 245.

∑ cosec²rπ/n, 209, 228.

binomial, 253.

calculus method, 128, 132 (ii).

definitions, 77, 189.

difference method, 127, 130.

products and, 228.

sh, sin, sinh, see ch, cos.

sin (A + B), 123.

Solution of Triangles, 1, 2, 19.

Submultiple angles,

cos θ, sin θ in terms of cos θ, 41.

cos θ, sin θ in terms of sin θ, 41-43.

cos θ in terms of cos θ, 43.

Subsidiary angle, 1.

Successive approximation, 82, 95.

Sum to infinity, 77, 190.

tan nθ, tan ∑θ, 172, 173.

tan x, 82.

tan^–1x, 88, 57.

Trigonometrical factors, 221.

Wallis’ limit for π, 223.