The London, Edinburgh and Dublin philosophical magazine and journal of science . point 0, and k, I those about any otherpair of rectangular axes through the same point, h being theproduct-coefficient about the same pair (i. e. the product ofinertia divided by the area of the section), wq have k2 = a2cos26 + b2sm20, (1) l2=a2sm2e + b2cos20, (2) h =02-52) sinfl cos 6, .... (3) where 6 is the angle KOA, considered positive when measuredfrom + OA towards + OB, and when the right angle from+ 0K to +0L is measured in this positive direction. First Method of Geometrically representing the aboveRelati
The London, Edinburgh and Dublin philosophical magazine and journal of science . point 0, and k, I those about any otherpair of rectangular axes through the same point, h being theproduct-coefficient about the same pair (i. e. the product ofinertia divided by the area of the section), wq have k2 = a2cos26 + b2sm20, (1) l2=a2sm2e + b2cos20, (2) h =02-52) sinfl cos 6, .... (3) where 6 is the angle KOA, considered positive when measuredfrom + OA towards + OB, and when the right angle from+ 0K to +0L is measured in this positive direction. First Method of Geometrically representing the aboveRelations. With diameter equal to Fig. 1. a + b describe a circlepassing through 0, butotherwise in any positionwhatever in the plane ofthe section, cutting theprincipal axes OA, OB inA, B respectively. Thismay becailed the gyration-circle at 0. On the diameter AB ofthe circle take a point P,such that PA = a, andYB = b. Then, if OK, OL are apair of rectangular axes, cutting the circle in K, L respectively,PK is the radius of gyration about OK;PL is the radius of gyration about OL ;. * Communicated by the Author. Phil. Mag. S. 5. Vol. 22. No. 138. Nov. 1886. 21 454 Mr. A. Lodge on a New Geometrical Representation of and twice the triangle KPL is the product-coefficient ahoutthe pair OK, OL. For, draw PM perpendicular to AK ; then PM = a cos 0,MK = 5 sin 0 ; hence PK2 = a2 cos2 0 + b2 sin2 0 = k2. Similarly, PL2 = Z2. Also LK (the base of the triangle KPL) = a + 6 ; and, if Sis the centre of the circle, SP = i(a— b), and the angleKSA=20, therefore the height of the triangle = i(a — b) sin 20= (a — b) sin 0cos 0 ; .*. twice area of triangle KPL = (a2 — b2) sin 0cos 0 = The sign of the product h is positive if the positive direc-tions of OK, OL include between them the axis of minimummoment (as in the figure), for in that case 0 is positive andless than a right angle, and a2—b2 is positive ; or 0 is nega-tive and less than a right angle, and a2 — b2 is
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Keywords: ., bookcentury1800, bookdecade1840, booksubjectscience, bookyear1840
