This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 Excerpt: ...hand, if l 1, the strain is a compression. EXERCISES 1. Repeat Ex. 1 of 2, drawing the auxiliary circle and constructing the diameters conjugate to the given diameters by application of Theorem 11. 2. Draw in pencil the asymptotes and a number of pairs of conjugate diameters, including the axes, of a rectangular ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 Excerpt: ...hand, if l 1, the strain is a compression. EXERCISES 1. Repeat Ex. 1 of 2, drawing the auxiliary circle and constructing the diameters conjugate to the given diameters by application of Theorem 11. 2. Draw in pencil the asymptotes and a number of pairs of conjugate diameters, including the axes, of a rectangular hyperbola. Construct in ink the lines into which the given lines are carried by the compression of ratio f which has an axis of the hyperbola as central line. What does the resulting figure represent? 3. Prove Th. 10, 6, by means of formulas (6 a) and (7) of the present paragraph. 8. Harmonic Division. Let P be a line-segment, and let Qi be one of its points. Then Qi divides PiP2 internally in a certain ratio, p (Ch. I, 6): QiP2 /i-Fig. 19 On PiP2 produced construct the point Q2 which divides PiP2 externally in the same ratio: The points Qi and Q2 are said to divide the segment PiJP2 harmonically; they divide PiP2 internally and externally in the same ratio: (V)-iQi _ Q2Pi. Qi'2 Q2P2 Let us start with / as given and trace the changes in Qi and Q2 as / varies. If / = 0, Qi and Q3 coincide in Pv As tt increases from 0 to 1, Qi moves to the right from Pi to the midpoint M of PiP2, and Q2 moves to the left and recedes indefinitely. If ix = 1, Qi is at M; but Q2 has disappeared. 7%2ts fAere 2s no point which, with the mid-point of a segment, divides the segment harmonically. As /j. increases from 1 without limit, Qi proceeds from M toward P2 as its limit, and Q2 appears again from the extreme right, continually moving in and approaching P2 as its limit. The proportion (1) may be written in the form: in PiQ, i _ QiP2. k' Q2A Q2P2' this new proportion says that Pi and P2 divide the segment QiQ2 harmonically. Thus we have the following t...
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