File:Graphical and mechanical computation (1918) (14764219694).jpg

Identifier: graphicalmechani01lipk (find matches)
Title: Graphical and mechanical computation
Year: 1918 (1910s)
Authors: Lipka, Joseph, 1882-1924
Subjects: Engineering Graphic methods
Publisher: New York : J. Wiley
Contributing Library: University of California Libraries
Digitizing Sponsor: Internet Archive

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Fig. 60a. 210 , _ 9000 If we Now let a = 10, anda^ a we have w^ — 2.1 w — g = o. The index line joining p = —2.1 and 2 = —9 cuts the curve in w = 2.42; hence w = 24.2 (Fig. 60b). We may similarly build an alignment chart for the quadratic equa-tion w^ + pw -(■ q = o. The method of construction of the curved axisfor this equation differs from the construction of the curved axis for thecubic only in that iv^ replaces li^, so that DD = z = —w^/(w + i)-Again we note that the points A (q = o), D (w = Wo), and E (p = —wq)must lie on a straight line, since these values of q, w, and p satisfy theequation w^ + pw -\- q = o. We may use this fact in constructing thepoints of the curv^e. The complete curve is drawn in Fig. 60b (curvemarked Q); by means of it we can find the positive roots of theequation. Example 3. Solve the equation ic^ — 5.15 7V + 2.42 = o. The indexline joining p = —5.15 and q = 2.42 cuts the curve m w = 0.52 andw = 4.65.
Text Appearing After Image:
112 alignment chart for solution of quadratic and cubic equations.Fig. 6o&. Art. 6o THE SOLUTION OF CUBIC AND QUADRATIC EQUATIONS II3 The complete cubic equation w^ + nw^ + pw + g = o may be trans-formed into the equation w^ + pw + 5 = o by the substitution w = w , or we may proceed to solve it directly by means of an alignment 3chart. In the construction of the curved axis we have AD=Zi = low and DD = 2 = - w -\- I I ^ \\ ( w^ \ \ w-\-i/ \ w-\-i/ In Fig. 60c, let the curve yl(2 correspond to the quadratic w- + pw -\- q = oand the curve A Co to the cubic w^ + pw -\- q = o (where w = o). Then DQ= - DCo= - DD = DCo + n • DQ. I w -(- I Thus starting at Co, we simply lay off, along DD, n times the fixed dis-tance DQ to arrive at the point Cn of the curve corresponding to the com-plete cubic equation. We thus rapidly layoff from Co in either direction a uniformscale with interval equal to DQ, and getthe points Ci, Ci, Cz, . . . , C-i, C_2, C-z corresponding to the curves w = I, 2, 3,

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