ON THE EQUILIBRIUM AND STRENGTH OF ELASTIC SUBSTANCES. 49 



mity will be half the versed sine of an 

 equal arc ia the circle of curvature at the 

 fixed point. 



The strain on each part is here equal to the weight of 

 the portion beyond it, acting at the end of a lever of half 

 ^s length : the curvature will therefore be as the square of 

 the distance from the extremity. And if the second fluxion 

 at the vertex be aa.fi, it will be every where (a— .r)*i.r: 



aaii 2arJi+x'ix ; the first fluxions of these quantities 



are aaxi and aaxi — ax'i+^i, and the fluents ^aV, and 

 ioV — ^<ur'+,!jX* ; or, vrhen x=a, lo'' and ^a' ; therefore 

 the depression is in this case half of the versed sine. 



328. Theorem. The height of the mo- 

 dulus of the elasticity of a bar, fixed at one 

 end, and depressed by its own weight, is half 

 as much more as the fourth power of the 

 length divided by the product of the square 

 ef the depth and the depression. 



The weight of the bar operates as if it were concentrated 

 at the distance of half the length, or as if it were reduced to 

 «ne half, acting at the extremity : we have therefore 



— for the length of a portion equivalent to the weight, and 

 a 



te hbm , 3e' . , , , . , Sc* 



— ^ -, whence mz:.-—j, and the hercht -77-. 



Ad 12c/ bh(t" * 2Md 



3«9. Theorem. The depression of the 

 middle of a bar supported at both ends, pro- 

 duced by its own weight, is five sixths of the 

 versed sine of half the equal arc in the circle 

 of least curvature. 



The curvature varies as aa — xr, and the second fluxion 

 is therefore represented by aaxi — xxxx, while that of the 

 "versed sine is aaii ; the first fluxions are aaxi and aoxi— 

 ft'i, and the fluents la'x'and ia'x' — ^!ji', or, when xzza; 

 ia*, and ,La', which are in the ratio of 6 to 5. 



330. Theorem. The height of the mo- 

 dulus of the elasticity 0+' a bar, supported at 

 both ends, is -^^ of the fourth power of the 

 length, divided by the product of the depres- 

 sion and the square of the depth. 

 . Forthestrainatthemiddleisequalto theeffectofthevreight 

 of one fourth of the bar acting on a lever of half the length 



, ^ J t 1. « 1 . ^^^ I'b/n 



(3 J 2 J ; and the radius of curvature there is r: — j and , 



'^~Thbd.' *"''"'* ''"S'^' 7^' sutstituung ! for/ 

 ■««L. II. 



Scholium. From an experiment made by Mr. Leslie 

 on a bar in the^e circumstances, the lieight of the modulus 

 of the elasticity of deal appears to be abuut 0,328,000 feet. 

 Chladni's observations on the sounds of fir wood, afford 

 very nearly the same result. 



331. Theorem. The weight under which 

 a vertical bar not fixed at tiio end, may be- 

 gin to bendj is to any weight laid on the 

 middle of the same bar, when supported at 

 the extremities in a horizontal position, 

 nearly in the ratio of t'4-J^ of the length to 

 the depression. 



For the weight laid on the bar being/, the pressure on 



/ « 



each fulcrum is — , and the length of the lever — , so that 

 2 2 



the weight of the modulus becomes 



iUd 



but the force. 



capable of keeping the column bent is ( — j. — or 



\ e / 12 



since e=:a, — ./::r.05i4— /. The effect of the weightof 



48a d 



the bar in the depression may be separately observed and . 

 deducted. 



332. Definition. The stiffness of bodies 

 is measured by their resistance at an equal 

 linear deviation from their natural position. 



333. Theorem. The stiffness of a beam; 

 is directly as its breadth, and as the cube of 

 its depth, and inversely as the cube of its> 

 length. 



Since TOZi—./ (326), and m varies as Ih, h being the 



breadth, I'dh varies as e'/, and/as , that is, when d , 



is given, as h, as P, and inversely as e'. 



334. Theorem, The direct cohesive or 

 repulsive strength of a body is in the joint ' 

 ratio of its primitive elasticity, of its tough- 

 ness, and the magnitude of its section. 



Since the force required to produce a given extension is 

 as the extension, where the elasticity is equal, the force at 

 the instant of breaking is as the extension which the body 

 ^11 bear without breaking, or as its toughness. And the 

 force of each panicle being equal, the whole force must be 

 as the number of the particles, or as the section. 



ScHOitUM. Though most natural substances appear 



