OF THE EQUILIBaiUM AND STBENGTH OF ELASTIC SUBSTANCES. 47 



this square becomes -77- , and the curvature varies directly 



as a, and as/, and inversely as *' : but since m vanes as b, 

 we must make the expression for the radius of curvature 



HJUL, which becomes ^when a—\l, and which varies as 

 12a/ 2/ 



i* directly, and aso and/ inversely. 



If the force be applied obliquely, its effect may be deter- 

 mined by finding the point at which it meets the perpendi- 

 cular to the axis, and resolving it into two parts : that which 

 is in the direction of this perpendicular will be counteracted 

 by'the lateral adhesion of the substance, the other will al- 

 ways produce the same curvature as if the force had been 

 originally in a direction parallel to the axis : but the place 

 of the point of indifference will be determined from the 

 point of intersection already mentioned, and when the force 

 becomes perpendicular to the column, the neutral point 

 will coincide with the axis. 



Scholium. If one surface of the column were incomr 

 pressible, and all the resistance of its strata were collected 

 in the other, the radius of curvature would evidently be 



*^'" , a being the distance from the incompressible side, 



which is ultimately 12 times as great as in the natural state 

 of an clastic substance. 



322. Theorem. The distance of the 

 point of greatest curvature of a prismatic 

 beam, from the line of direction of the force, 

 is twice tlie versed sine of that arc of the 

 circle of greatest curvature, of which the ex- 

 tremity is parallel to that of the beam. 



Supposing the curve, into which the beam is bent, to be 

 described vnth an equable angular velocity, its fluxion will 

 be directly as the radius of curvature, or inversely as a, the 

 distance of the force from the axis of the beam ; this we 

 oiay stiU call a at the point of greatest curvature, and y 

 elsewhere, the corresponding arc of the circle of curvature 



being x ; then the fluxion of the curve will be_l ; but this 



y 



fluxion is to y as the radius r to the sine of the angle or arc 



■ ^ a': sin.z , fsin.i)s . ... , 



1, or ^y , but m', v bemg the versed 



y r T 



sine of the ate z, Zf:yy::za- , and yy:z:b^iav, t being a 

 constant quantity : when yZZa, u^o, and aazzb, therefore 

 yy^zaa — 2a«, and when y—0, aazz'lav, and az:2u. 



Scholium. When the force is longitudinal, and the 

 curvature inconsiderable, the form coincides with the har- 

 monic curve, the curvature being proportional to the dis- 

 tance from the axis : and the distance of the point of in- ' 



difference from the axis becomes the secant of an arc pro- 

 portional to the distance from the middle of the column. 



323. Theorem. If a beam is naturally 

 of the form wliicii a prismatic beam would 

 acquire, if it were slightly bent by a longitudi- 

 nal force, calling its depth, b,\\s length, f, the 

 circumference of a circle of wiiich the dia- 

 meter is unity, c, the weight of the modulus 

 of elasticity, m, the natural deviation from 

 the rectilinear form, d, and a force applied at 

 the extremities of the axis,/, tiie total devia- 

 tion from the rectilinear form will be a = 



hhccdm 

 bbicm—l2eej" 



The form being originally a harmonic curve, the curva^ 

 ture and length of the ordinate added at each point by the 

 action of the force will also be equal to those of a harmonic 

 curve, of which the vertical radius of curvature must be 



■ ^ and the basis the length of the beam; but tlie 



12rt/ 



vertical ordinate of the harmonic curve is a third propor- 

 tional to its radius of curvature and that of the figure of sines- 



on the same basis, which in this case would be — , the ad- 



c 



ditional vertical ordinate must therefore be _ . — :., and 



cc bbm 



this added to the deviation d, must become equal to a, and' 



ee ^laf abbccm — llaeef 



— , — --^d— ^ and o 



cc bbm bbcon 



, ee \-lttf 

 Zd+—-fL, 

 cc bbm 



blcrdm 



bbccm — lieef 



ScHOLiuw. It appears from this formula, that when 

 the other quantities remain unaltered, a varies in propor- 

 tion to d, and if rfr:n, the beam cannot be retained in a 

 slate of inflection, while the denominator of the fraction ■ 

 remains a finite quantity : but when bbccm'iZZlieeJ, a be- 

 comes infinite, whatever may be the magnitude of d, and 

 the force will overpower the beam, or will at least cause it 

 to bend so much as to derange the operation of the forces 



concerned. In this case / — I — i — , .8225 — m, which 

 \ e / 12 ee 



is the force capable of holding the beam in equilibrium in 



any inconsiderable degree of curvature. Hence the modulus 



being known for any substance, we may determine at once 



the weight which a given bar nearly straight is capable of 



supporting. For instance, in fir wood, supposing its height 



10,000,000 feet, a bar an inch square and ten feet long may 



