128 Mr. K. Mallet on the Origin and Mechanism of 



is, to the lengths of the lines of fracture. Referring now to the 

 diagram (tig. 1), in which the three possible forms of subdivi- 

 sion are shown in two adjacent triangles, squares, and hexagons, 

 the areas of all of which are equal, let rupture be supposed 

 to take place between two adjacent areas along the lines of con- 

 tact Im of two adjacent figures, the rending forces proportionate 

 to the areas may be supposed situated at the centre of gravity 

 of A and B, and to act in the directions (along the line Im) of 

 the short convergent arrows. The resistance of the material to 

 rupture is proportional to the perimeter of the figure ; or, as 

 every side of every figure is opposed to a similar side of another 

 adjacent figure, it is proportional to Im, or to a third of the 

 total perimeter of the triangle, to one fourth of the perimeter of 

 the square, and to one sixth that of the hexagon; and the re- 

 sistance to rupture along the lines / m is in a direction perpen- 

 dicular to the same. Though the total rupturing force, being 

 proportional to the area, is equal in all the figures, the effective 

 rupturing force differs in all three, being dependent upon the 

 relative obliquity with which the tensile forces between A and 

 B act along the line Im — in other words, upon the integrals 

 upon all possible angles of direction along Im between the direct 

 pull AB and the most oblique one I Km. In other words, if/ 

 be the direct pull between A and B and 6 be the angle at A 

 between that line and that in the direction of any oblique force 

 indicated by any one of the arrows, /"cos 6 is the rending force 

 in the latter direction. Integrating graphically and approximately 

 the total rending effort along and perpendicular to the line Im in 

 the three respective figures by dividing the angle which it sub- 

 tends in each respectively (viz. 120° in the triangle, 90° in the 

 square, and 60° in the hexagon) into an equal number of equal 

 parts, we find that the total effective effort to produce rupture 

 along the line Im in the three figures is proportional to the fol- 

 lowing numbers : — 



Equilateral triangle . . 6*564 



Square 8'497 



Hexagon 10261 



or to the numbers 100, 129, 157. The rending effort is there- 

 fore much the greatest in the hexagon. But the resistance to 

 rending is, as has been already shown, proportional in each 

 figure to its perimeter or to the length Im of one of its sides 

 along which rupture occurs ; and the perimeters respectively for 

 equal areas are as follows : — 



Equilateral triangle . . 45 6 



Square 40-0 



Hexagon 37'224 



