An Elementary Treatise on Curve Tracing Book

Chapter I.	Introductory Theorems. Definitions. Tracing by Points. Symmetry
Art 3-4	Theorems for working	2
Art 5	Definitions	5
Art 6	Curves traced by points	6
Art 7	Symmetry	7
	Examples I	8
Chapter II.	Orders of Small Quantities. Forms of Parabolic Curves Near the Origin. Cusps. Tangents to Curves. Curvature
Art 9-10	Orders of small quantities	9
Art 11-12	Standards of vanishing quantities	10
Art 13-14	Graphic construction of parabolic curves	12
Art 15	Tracing of y[superscript m]=cx[superscript n] near the origin	13
Art 16	Shape at any point	14
Art 17	Example of a point of inflexion	15
Art 18	Algebraical representation of a cusp	15
Art 19-20	Tangent to a curve	16
Art 21	Measure of curvature	17
Art 22	Circle of curvature	18
	Examples II	18
Chapter III.	Forms of Parabolic Curves at an Infinite Distance. Examples of Tracing Curves. Trigonometrical Curves. Illustrations of Theory of Equations. Rules for Approximation
Art 26-28	Forms of parabolic curves at a great distance from the origin	20
Art 29-31	Examples of finding a tangent to a curve, where it is parallel to an axis, and the position of a point of inflexion	21
Art 32-33	Examples of tracing curves in which y can be expressed in terms of x explicitly	22
Art 35-38	Curves representing changes of trigometrical functions	28
Art 39-41	Illustrations of theory of equations	29
Art 42-45	Condition for a point of inflexion	31
Art 47-49	Graphic solution of equations	32
Art 51	Working formulae for approximation	33
Art 52	Illustrations of methods of approximation	34
	Examples III	36
Chapter IV.	Forms of Curves in the Neighbourhood of the Origin. Simple Tangents. Direction and Amount of Curvature. Multiple Points of Two Branches. Curvature of Branches at Multiple Points. Multiple Points of Higher Orders
Art 54	Forms of curves near the origin	39
Art 55-57	Simple tangent	39
Art 58-60, 62	Conic of curvature	40
Art 60, 61	Circle and diameter of curvature	41
Art 63	Examples of curvature	42
Art 64	Tangent to u[subscript 1] + u[subscript 1]v[subscript 1] + u[subscript 3] + ... = 0	44
Art 66	Tangents to multiple points of two branches	45
Art 67-69	Form of v[subscript 1]w[subscript 1] + u[subscript 3] + ... = 0	45
Art 70, 71	Form of v[subscript 1]w[subscript 1] + u[subscript 4] + ... = 0	48
Art 72, 73	Form of v[subscript 1]w[subscript 1] + v[subscript 1]v[subscript 2] + u[subscript 4] + ... = 0	49
Art 74, 75	Form of v[subscript 1 superscript 2] + u[subscript 3] + ... = 0	51
Art 76, 77	Form of v[subscript 1 superscript 2] + v[subscript 1]v[subscript 2] + u[subscript 4] + ... = 0	51
Art 78, 79	Form of v[subscript 1 superscript 2] + w[subscript 1 superscript 2] + u[subscript 3] + ... = 0	53
Art 80, 81	Curvature of branches at multiple points	54
Art 82, 83	Multiple points of higher orders	54
	Examples IV	56
Chapter V.	Forms of Branches Whose Tangents at the Origin are the Coordinate Axes
Art 84-88	Forms when x and y are of different orders of magnitude	58
Art 89	Tentative methods	61
Art 90	Rules for rejecting terms	61
Art 91	Worked examples	62
	Examples V	66
Chapter VI.	Asymptotes. Points of Intersection at an Infinite Distance. Asymptotes Parallel to the Axes
Art 94, 95	Rectilinear and curvilinear asymptotes	68
Art 96-98	Intersection of curves with their asymptotes	69
Art 99-102	Comparison of singular asymptotes with tangents at points of inflexion and multiple points at a finite distance	71
Art 103, 104	Worked example	73
Art 106, 107	Determination of asymptotes by points of intersection at an infinite distance	74
Art 107-110	Worked examples	74
Art 112	Method by approximation	79
Art 113	Cases of x alone infinite	80
Art 114	Side on which the curve lies	80
Art 115, 116	Varieties in the determination of asymptotes which are parallel to the axes	81
Art 117	Examples of asymptotes	82
	Examples VI	87
Chapter VII.	Asymptotes Not Parallel to the Axes. Asymptotes to Homogeneous Curves
Art 119	Simple examples of asymptotes not parallel to the axes	88
Art 120	Illustration by multiple points at a finite distance	90
Art 121	General statement of the method of finding by approximation those asymptotes which are not parallel to either axis	90
Art 122-124	Side on which the curve lies	91
Art 125	Case of a parabolic asymptote	92
Art 126	Investigation of a proper parabolic asymptote	93
Art 127	Examples of parabolic asymptotes	93
Art 128-130	Parallel rectilinear asymptotes	95
Art 131	Direct method of expansion	97
Art 132	Observations relating to the side on which the curve lies	97
Art 134	Homogeneous curves and their asymptotes	100
	Examples VII	106
Chapter VIII.	Curvilinear Asymptotes
Art 136-138	Infinite branches when x and y are not of the same order of magnitude	107
	Examples VIII	115
Chapter IX.	The Analytical Triangle. Properties of the Analytical Triangle
Art 140-143	Newton's parallelogram and De Gua's triangle	117
Art 144	Modified form in which the triangle is employed in this work	119
Art 145-147	Statement and proof of properties of the triangle	119
Art 148	Examples of the use of the triangle	122
	Examples IX	131
Chapter X.	Singular Points. Division into Compartments. Special Curve of the Fourth Degree
Art 157-160	Degeneration of curves of the form y[superscript m]=cx[superscript n]	135
Art 161, 162	General conditions for singular points	137
Art 163	Examples of singular points	138
Art 164-166	Assistance derived from compartments	143
Art 167, 168	Methods of detecting isolated portions of a curve	149
Art 170-192	Symmetrical curve of the fourth degree of the form [characters not reproducible]	152
	Examples X	165
Chapter XI.	Systematic Tracing of Curves. Repeating Curves
Art 194	Rules for systematic tracing	167
Art 195-197	Worked examples	168
Art 199-202	Class of repeating curves	177
	Examples XI	184
Chapter XII.	Inverse Process. Determination of the Equation of a Given Curve
Art 210-212	Difficulties of the inverse process	186
Art 213-215	First steps towards the solution of the problem	187
Art 217-222	Method of making use of the analytical triangle	193
Art 223	Use of the rule of signs	195
Art 225	Examples of use of rule of signs	195
Art 226-231	Use of partial curves and compartments	200
	Examples XII	202
	Plates	203
	Classified List of the Curves Discussed	239
	Index	245