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LIII. New Geometrical Representation of Moments and Pro- 

 ducts of Inertia in a Plane Section; and also of the Re- 

 lations between Stresses and Strains in two Dimensions. 

 By Alfked Lodge, M.A., Coopers Hill, Staines *. 



THE object of the first part of this paper is to give two 

 methods by which the connection between the moments 

 and products of inertia about pairs of rectangular axes through 

 a point may be represented by means of a circle, without the 

 necessity of drawing the ellipse of inertia. 



If a, b are the radii of gyration about the principal axes 

 OA, OB at the given point 0, and k, I those about any other 

 pair of rectangular axes through the same point, h being the 

 product-coefficient about the same pair (i. e. the product of 

 inertia divided by the area of the section), wq have 



k 2 = a 2 cos 2 6 + b 2 sm 2 0, (1) 



l 2 =a 2 sm 2 e + b 2 cos 2 0, (2) 



h =0 2 -5 2 ) sinfl cos 6, .... (3) 



where 6 is the angle KOA, considered positive when measured 

 from + OA towards + OB, and when the right angle from 

 + 0K to +0L is measured in this positive direction. 



First Method of Geometrically representing the above 

 Relations. 



With diameter equal to Fig. 1. 



a + b describe a circle 

 passing through 0, but 

 otherwise in any position 

 whatever in the plane of 

 the section, cutting the 

 principal axes OA, OB in 

 A, B respectively. This 

 may becailed the gyration- 

 circle at 0. 



On the diameter AB of 

 the circle take a point P, 

 such that PA = a, and 

 YB = b. 



Then, if OK, OL are a 

 pair 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. 



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