Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 edition. Excerpt: ...MN the initial position of the line. Fig. 64. "Step off" any small equal arcs on the circumference of C as AB, CD, DE, etc. Draw tangents at the points of division and beginning with A stepoff, successively 1, 2, 3, 4, etc., times the distance AB on the tangent lines. The resulting points will determine an involute. Any curve whatever will produce an involute in this way, but the circle is most commonly used. A gear tooth is made up of cycloid, evolute, and circular arc in varying proportions. SPIRALS. Art. 118. A spiral is described by a point receding, according to some fixed law, along a straight line that revolves about one of its points. There are a number of spirals, one of which will illustrate this type of curve. The revolving line is called the radius vector and the angle it makes, in any position, with the initial line, is called the vectorial angle. The hyperbolic spiral is the curve generated by a point, which moves so that the product of radius vector and vectorial angle is constant. Fig. 65. Calling the radius vector, r; the vectorial angle 0 and the constant C, we have by definition, rd=C. 11 To construct it when C = 11, then r.. Make a table of values for r, as follows; When ELEMENTARY CALCULUS. CHAPTER I. FUNDAMENTAL PRINCIPLES. Art. I. Variables and constants. Suppose we wish to plot a curve, corresponding to the relation y = x3 + 2 x2--5 x--6; and for this purpose assign to x certain arbitrary values, calculating from these the corresponding and dependent values of y. Now in such a case both x and y are variable quantities, x being called an independent, and y a dependent variable. In general: A Variable is a quantity which is subject to continual change of value, while an Independent Variable is supposed to...