46 OF THE EQUILIBRIUM AND STRENGTH OF ELASTIC SUBSTANCES. 



30 4p 



— , therefore the whole effect is — , which repreients the 



aa aa 



attraction of the whole surface at the distance a. 



SECTION IX, OF THE EQUILIBRIUM' AND 

 STRENGTH OF ELASTIC SUBSTANCES. 



318. Definition. A substance perfectly 

 elastic is initially extended and compressed 

 in equal degrees by equal forces, and pro- 

 portionally by proportional forces. 



319. Definition. The modulus of the 

 elasticity of any substance is a column of 

 tiie same substance, capable of producing a 

 pressure on its base which is to the weight 

 causing a certain degree of compression, 

 as the length of the substance is to the dimi- 

 nution of its length. 



320. Theorem. When a force is applied 

 to an elastic column, of a rectangular pris- 

 matic form, in a direction parallel to the axis, 

 the parts nearest to the line of direction of 

 the force exert a resistance in an opposite 

 direction; those particles, which are at a 

 distance beyond the axis, equal to a third 

 proportional to the depth and twelve times 

 the distance of the line of direction of the 

 force, remain in their natural state ; and the 

 parts beyond them act in the direction of the 

 force. 



The forces of repulsion and cohesion are initially propor- 

 tional to the compression or extension of the strata, and 

 these to their ilistance from the point of indifference : the 

 fojces may therefore be represented by the weight of two 

 triangles, formed by the intersection of two lines in the 

 point of indifference ; and their actions may be considered 

 OS concentrated in the centres of gravity of the trianf,les, 

 which are at the distance of two thirds of tlie length of each 

 from the vertex, and at the distanceof two thirds of the depth 

 from each other. This distance constitutes one arm of a lever, 

 which is of constant length, while the distance of the line of 

 direction of the force from the centre of gravity of the nearest 

 triangle constitutes the other arm ; and calling the distance 

 ei the line of direction of the lo.ce from the axis, a, and the 



depth, b, the length of this arm, on the supposition that 



the point of indifference is at the assigned distance, will be 



lb f hb \ lb 

 a-\ IIt^H l.ora-1 \l, that of the con- 



slant arm being \b. The cohesive and repulsive force* 

 lb hb 



must therefore be as o+- 



3(ia 



-i'to°+-^+T^ since 



that vthich serves as the fulcrum of the lever must bear a 

 force equal to the sum of the two forces applied at the ends, 

 which are proportional to the opposite arms of the lever ; or 

 as Z6aa — laai+td toSBaa+iaoi+M, that is, as {6a — i)« 

 to (6n+i>)' : but these forces arc actually as the squares of 

 the sides of the similar triangles which represent them, 



that is, as ^ \b- ~) to(^ '-1 +-^^ , or as (6a— i)' 



to (Ba+t)', which is the ratio required : there will there- 

 fore be an equilibrium under the circumstauces of the pro- 

 position. 



321. Theorem. The weight of the mo- 

 dulus of the elasticity of a column being »», 

 a weight bending it in any mannery, the dis- 

 tance of the line of its application from any 

 point of the axis, a, and the depth of the 

 column, b, the radius of curvature will be 

 bbm 



Supposing first the force to act longitudinally, and azi. 



^, the point of indifference will be in the remoter surface 



of the column, and the compression or extension of the 



nearer surface will be twice as great as if the force had been 



applied equally to all the strata; and will therefore be to 



the length of any portion as ifro m ; but as this distance is 



to the length, so is the depth to the radius of curvature, or 



bm 

 ftf : m : : b : — , which is the radius of curvature when 



azz^l. But when a varies, the curvature will vary i the 



same ratio ; for the curvature is proportional to the angle 



of the triangles representing the forces, and the angle of 



either triangle to its area divided by the square of its length; 



but the force exerted by the remoter part of the column is 



II 



to/ as 0-1 ii to ib, or as (<Ja — M' to 24o?', and is 



•' sea ' ' ' ^ ' 



f 

 equal to ■ lea — J)', but the square of the side of 

 2*ao 



/ bb \i 



the corresponding triangle is I {b TTT) t °' (*" — *)*- 



(b \« 

 — • ) , consequently the force, or ths area, divided bj 

 laa / 



