598 REPORT — 1893. 



the sum of a series of second products. Tlie figure already used (fig. 12) 

 may be employed to indicate the process. In this figure the first products 

 have been obtained, and by taking a new pole Pj at a distance of unity, 

 and drawing the rays as shown, and then drawing the parallels precisely 

 in the sa,me way as in the first case, a number of values of the form 

 OA"^ . K are obtained, each of which is called the ' moment of inertia ' of the 

 given segment about that point,' so that the length of 0' A4" becomes 



S0A2 . K=I=Moment of Inertia. 



It may be required to obtain a series of the form 



S{(OA.K)0'A'}, 



where the values of OA and O'A' both change, in which case a graphical 

 solution is also readily obtained. 



The construction for continuous products enables the mean value to 

 be found of OA (say OA,.), such that 



This is done by producing the two extreme lines in the second derived 

 figure, which were drawn parallel to those of the first figure, until they 

 intersect, and through the point of intersection drawing a parallel to the 

 line BCD, and the intersection of the line with the line OAj A2 . . . 

 giving the required distance. This is shown in fig. 12. Now it does not 

 matter what distance O is taken along the line Aj A2 A3 . . . ; the position 

 A,, relatively to Ai, A2, A3, ... is always the same. 



Now the actual position of the segments in the direction of their own 

 length has nothing to do with the result, and wherever O is taken, so 

 long as the perpendicular distance to the segments is unaltered, the 

 distance A,, of the perpendicular upon the direction of the segments 

 through O remains the same. Moreover, the sum of the products of the 

 segments into their distances from A,, (having due regard to their sign) 

 is zero. 



Suppose that the number of segments is infinitely great, and that 

 they are infinitely close together, so that their extreme points form the 

 locus of the boundary of a plane figure ; then the sum of the products about 

 any point in the line through A,., (fig. 15), parallel to the direction of the 

 segments, is zero. Next, suppose the same boundary to be formed by the 

 extreme points of a similar second series of parallel segments, taken in 

 some other direction. The same will hold with regard to the line through 

 some point A,., for the second series, and parallel to them. The inter- 

 section of these two lines gives a point G about which the sum of the 

 products of every series of parallel segments whose extreme points lie on 

 the boundary of the figure is zero. This point is called by Cremona and 

 others the ' centroid ' of the area in question, and its determination for 

 surfaces of various kinds is a matter of great practical importance. 



The way in which the matter has been approached, though not perhaps 

 the simplest from the point of view of application in mechanics, indicates 

 at once the graphical loethod of determining centroids. Thus any plane 



' This term is quite meaningless for the most usual applications in engineerings 

 and might be replaced by some other, as it presents a difficulty to those who hav& 

 not had a mathematical training. 



