21 



(liit'i'tioiis. lie supposed tlirci' axes, one lyaiullo.l to tlu' (ihi-rs, tlic second in tli(^ direi'fion oC the 

 radius, tlie tliiid tangential to the annual iin,i;-. He (bund tliat if tlie resistance against cross 

 bending in the direction to the tangent is taken as unity, the resistance in the direction of the 

 radius is 2.25, that in the direction of th(i fiber is 10. (Menioires de l'Aca(h''niie des Sciences, 1830.) 

 The same question was discussed by Wiieatstone in the Philosophical Transactions, in 1833. 

 He says: 



^\^l(!n a plate is cut out so that tlie fibers run parallel to one of the siile faces the axes of the j,'reatest aud 

 smallest elasticity are rectauRular to each other ami parallel to the sides. If the plate has the shape of a rectangle, 

 the sides of which are inversely pro]iortional to the scjuares of their resistance to ben<Iins. both kinds of vibrations 

 parallel to the sides, although these may vary in length, become synchronous, and its consistency furnishes a 

 resulting figure with lines parallel to the diagonal. 



Llcnce, by tiuding experimentally the relative length that these sides should have, the relation 

 of the coeHicients of elasticity in two directions perpendicular to each other may be found. 



I'ONCELET, in his Mecaniiiue Industiiel,18:>9, enters into a very detailed discussion on the elas- 

 ticity of timbers and especially ou exiierimeuts into the lesistancc^ to torsional forces. He derived 

 from the experiments of Minard, Desormes, and Ardent the conclusion that for the first load the 

 elongations are ])roportional to the straifiing forces, and he calculates from these elongations the 

 coeflicieuts of elasticity. 



According to Eaton Hodgkinson (Combes, Exploitation des Mines), a shortetiing by 0.0027 of 

 the original letigth of a rectatigular (tiot bent) prism alters the elasticity considerably. 



Hacien (Poggendorfs Aniialeti, Vol. LViii, p. 12."j) has itivestigated the elasticity of several 

 kinds of timber by crossbending of sticks, which were cut in the one case in the direction of the 

 fiber, and ifi the other case perpendicularly to the same, and did not find atiy great dilference 

 betweefi the heart and sap wood; but he lecognized that the coefficient of elasticity decreases con- 

 siderably when the wood contains much water. 



In tlie year 1845 two Italian physicists, Paccinotti atid Pbki (II Cifuento, Vol. in), ])ublished 

 a very precise and detailed investigatiofi into the elasticity of titnbers, in which the various 

 methods for the determitiatiofi of the coefficient of elasticity were coini)ared and their value 

 examined. They operated according to the three methods of tension, flexion, atid torsioti, with 

 cubic blocks of 27 to 30 millimeters side. Iti the experiments witli flexion they employed dif- 

 ferent methods of fastening or support. They measured the elastic as well as the ])ermaiiefit 

 elongations, the augles of torsion, the ordinates corresponding to the different points of the stick 

 during its deflection with ati increasing load. In the second part of their work, Paccinotti and 

 Peri cofuiiare the numerical results of their experifnents with the figures which were calculated 

 by the ktiown formulas, aud attempt to find a relation between the density and the coeiUcient of 

 elasticity for the timbers experituented ujiou. 



They arrived at the following conclusions: 



(1) The elasticity allows in th<^ <litferent parts of the wood cliangcs in diiuiiusioii, which are in iiroportiou not 

 only to tlie first load but also to those whi<-li comi^ near that load wliich produces rupl-nre, provided care is taken to 

 ditferentiate from the elastic changes those permanent ones which are due to the softness of the material or to the 

 continuity of the load. 



(2) The curves of dellection wliich tlio sticks assume when fastened securely at one <iiil dili'cr uiiiler otherwise 

 e()iial conditions from those wlii<-li the same sticks fm-m when botli ends are supported ; this must be ascribed to the 

 reai-tion of the liber in the two opposite branches foriiK^d in the latter method of sujipori . The same theory, however, 

 may servo for the ]>nrpose of depriving the two kiiiils of curves, jirovided tliat in tlie iiitc-gial ion of the respective dif- 

 ferential cijiiations care is taken to projicrly determine the sonstant (the value of wliich depends on the degree of 

 the iuvarialillily of tlie fastening or insertion of the stic-ks (^xl>erillleuted upon). 



(3) The ditt'ercnccs whicli ajipear in tlu^ deteriiiiiiatiou of the coeflicieut of elasticity disappear almost entirely, 



if by this expression is understood tlic iiuoticiit ''j^,^.. i" which K denotes the common term for tlic (■oefUcieut of 



elasticity and O the specific gravity. 



(4) The coefficient of elasticity Ei, although there are some differences in the various species of wood, is in gen- 

 eral ciiual to 2,000 for the stiuaro millimeter of transverse cut. 



(.')) The coefficient of elasticity can be determined not only by tension but also by flexion and torsion, but with 

 these ditferent methods dill'erent values are founil, and to reduce these to thesame value it will be necessary in each 

 case to deteriuiue a constant coellicii^nt dcpeiidiMit ii|ion thi^ metliod of operation. 



(ti) The easiest metliod for the dctcriiuiiatioii of the coeflii^ieiit of elasticity is that in which Mie stick is sup- 

 ported at both ends and is loaded at equal distance from the two points of siipiiort (center). 



