Theory of Elasticity. 271 



of a chronometer, where two metals of different expansibility 

 in combination serve to correct the normal effects of change of 

 temperature. As a parallel to the effects of unequal heating 

 we have the effects of unequal swelling due to moisture. 

 Sometimes, as in the case of the windows, different parts are 

 differently affected ; sometimes, as with wood and rope, the 

 effect is different in different directions. Then we have the 

 effects of the cold working of material, as the cold rolling of 

 plates, the drawing of wire, the spinning of ductile metals, 

 shearing, punching, bending and twisting. Most metals, if 

 subjected to such operations and not relieved of the primary 

 stresses thus engendered, by subsequently annealing them, are 

 much less trustworthy under stress than before. And lastly 

 we may note the consequences of straining up parts of built- 

 up objects in their construction, as the tightening of nuts, 

 screws and wedges. Frequently most serious and objectionable 

 strains are thus introduced and added to all others. 



When we have a body supporting, — resisting, applied 

 forces, — loads ; what we generally wish to know is how well 

 able is that body to perform its duties, what are the actual 

 stresses and strains within it ? We know, if the body be at 

 rest, that the stresses must be in equilibrium among them- 

 selves and with the bodily forces (as weight, magnetism, etc.) 

 and the surface tractions. We know that on any complete sec- 

 tion of the body the resultant must be equal and opposite to 

 the resultant of the bodily forces and surface tractions acting 

 on the part of the body cut off by the section. Thus, when 

 the section becomes sufficiently small and circumscribes an 

 elementary portion of the body, we obtain as consequence 

 three general differential (interior) equations of equilibrium 

 and three general ordinary (surface) equations of equilibrium.* 



* Equations limiting actual stresses. 



For interior points. 





For surface points. 



dP dU dT 







- — + - — + - — = — pX 







ox oy oz 





IP+mU+nT =f] 



dx oy dz 



-(1) 



1 

 lU+mQ + nS = G K 2 ) 









IT +m~S + nR=H j 



6T dS 6B „ 

 1 — + i — + : — = — P z 









ox oy oz 







where with a system of rectangular coordinates xyz the quantities PQR are the 

 actually existing and complete intensities of the normal components of the stress 

 in the planes of x, y and z respectively ; "STUare the intensities of the tangential 

 components of the stress in the planes of y and z, 2 and x, x and y perpendicular 

 to x, y and z respectively ; p is the density and X Y Z the bodily forces per unit 

 volume at the point (xyz)) Imn are the direction cosines of the perpendicular to 

 the bounding surface at any point and FGH are the intensities of the surface 

 tractions at that point in the directions x, y and z respectively. 



