ON GKAPHIC METHODS IN MECHANICAL SCIENCE. 



599 



figure may be divided into parallel strips and segments, representing the 

 areas drawn through the centre of each ; the line containing the centroid 

 is then found. This operation is repeated with parallels in some other 

 direction, and the intersection of the two lines containing the centroid 

 gives the point required. 



Fig. 15. 



There are numerous propositions relating to figures of difierent known 

 forms given by Culmann and others, such as for the triangle, trapezium, 

 sector of a circle, segment of a parabola, &c., but the method of derived 

 figures enables the centroid of any plane area to be obtained. 



Segments representing volumes and the centroids of solids can 

 be found by a similar process, just in the same way as the centroid is 

 found for a series of segments when the first products are taken. 



A point A,, corresponding to the centroid, can be found in the per- 

 pendicular from any point when the second products are taken ; but it 

 may be remarked that 



(1) The square of the distances being taken, the sum of the products 

 about the point can never be zero. 



(2) The point Aj changes for every change in distance of the point O. 

 The determination of the distance, which is called the radius of 



gyration of this point from O, can be graphically obtained, and the 

 method is explained in Culmann's work, and by Chalmers and others. 



The segments have hitherto been drawn in the plane of the paper, but 

 there is no necessity to do this ; indeed, the cases which occur when 

 segments are multiplied are usually such that one series of segments is 

 taken as acting at right angles to the plane of the paper, their magnitude 

 being represented by an area, so that if an area is divided into a number 

 of equal parts the segments are equal, and equal in number to the 

 divisions. If the segment does not act at right angles to the plane, then 

 the intersection with the plane will give the shortest distance of the 

 segment fi'om it, and therefore the length of the second segment by which 

 it is to be multiplied. 



