604 



REPORT 1893. 



30 that tbe derived radius of gyration is the geometrical mean of the 

 distances »j and ri^. 



The geometrical property just given may be at once applied to another 

 very important purpose, viz., to find the mean distance from a line which, 

 multiplied by the sum of a series of segments whose magnitude is pro- 

 portional to their distance from the line '"i ques^'on, gives a result f qual 

 to the sum of the products of each segment into its distance from the 

 line. This is really the problem of finding the centre of pressure in a 

 surface over which the intensity of pressure varies nniformly, or the 

 centre of resistance of a bent beam, the intensity of stress varying 

 nniformly at the skin. 



Fig. 16. 



Let the area, fig. 16, be that of the surface in question, the distance of 

 every point of which from the tangent A B being a measure of the seg- 

 ment , which has to be multiplied into that distance. The sum of the 

 squares of the distances is the quantity required, and is thus found. Find 

 the central ellipse of the area, and through the centre draw the diameter 

 O C conjugate to the diameter parallel to A B, intersecting A B in C and 

 the ellipse in D. "From C draw the tangents C E and C F. Join E F, 

 intersecang O C in M. Take ON=OM in C O produced; then Nis the 

 required point. The point in question is the antipole of the line A B 

 relatively to the central ellipse of the given centre. 



If now for every position of the tangent A B round the given area, 

 antipoles are found, a curve is obtained which is called — in German, 

 liern ; in Fi'ench, noyau ; and in English kern, Icernel, core, and heart. 

 The first of these seems the best term. 



It has already been shown how the sum of a number of products may 

 be obtained for any conditions of sign, and in certain special cases it is 

 necessary to have a graphical statement of the result which one series of 

 the two quantities multiplied together varies between certain limits. 

 Bending moment diagrams, deflection curves, and moment of inertia 

 diagrams are examples of such graphical statements. 



The Sum of the Products of Nok-pakallel Segments. 



Hitherto the products of parallel segments have been dealt with, but 

 it may also be required to find the sum of a series of products in which 

 the segments representing different magnitudes of the same kind have 



