. Graphical and mechanical computation . in 11 1—r 103876 S 4 3 (u) \I, Fig. 21c. angle. From any point P draw PIi, Ph, Ph perpendicular to OX, OYand OZ respectively, cutting OX in A, OY in B, OZ in C. Let Ph alsocut OZ in D, and draw AE perpendicular to OD, cutting OZ in E. Then OC = OE -f- ED +DC - OA COS450 + AD cos 450 + DP cos 450 = 9A+^.. V2 Art. 21 HEXAGONAL CHARTS 4* Thus, if the axes OX, OY, OZ carry the scales x = mf(u), y = mF(v),z = —=(w) respectively, then the three perpendiculars from any point V2 to these axes will cut them so that m nm+mm m + = V2 V2 thus, any equation of this
. Graphical and mechanical computation . in 11 1—r 103876 S 4 3 (u) \I, Fig. 21c. angle. From any point P draw PIi, Ph, Ph perpendicular to OX, OYand OZ respectively, cutting OX in A, OY in B, OZ in C. Let Ph alsocut OZ in D, and draw AE perpendicular to OD, cutting OZ in E. Then OC = OE -f- ED +DC - OA COS450 + AD cos 450 + DP cos 450 = 9A+^.. V2 Art. 21 HEXAGONAL CHARTS 4* Thus, if the axes OX, OY, OZ carry the scales x = mf(u), y = mF(v),z = —=(w) respectively, then the three perpendiculars from any point V2 to these axes will cut them so that m nm+mm m + = V2 V2 thus, any equation of this type may be represented by three scales. Theindices I\, h, h may be drawn on a transparent sheet and this sheet is. Fig. over the paper keeping the indices perpendicular to the 2ic charts the equation —U- = — by this method. U V IV We can choose the modulus on OZ the same as the moduli for OXand OY by following the construction illustrated in Fig. 216. Here OXand OY cut at an angle of 1200 and OZ bisects this angle. Join OP andlet angle COP = a. Then 42 NETWORK OF SCALES Chap. II OA = OP cos (6o° + a) = OP (cos 6o° cos a — sin 6o° sin a).OB = OP cos (6o° - a) = OP (cos 6o° cos a + sin 6o° sin a)... OA +0B = 2 OP cos 6o° cos a = OP cos a = OC. Thus our three scales are x = mf(u), y = mF(y), z = m4>{w). Fig. 2\dcharts the equation nv = w or log u + log v = log w by this method. EXERCISES. i. Represent the equation v = u3 by a straight line, using natural scales. 2 Represent the equations (a) 2 u2 + 3 v2 = 6, (b) u2 — 3 w2 = I, (c) w2 -f- ^2 = 4by straight lines, using natural scales, and find graphically the simultaneous solutions ofthe three equations taken in p
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