722 PROFESSOR MACQUORN RANKINE ON THE DECOMPOSITION OF 



body's surface, left by the third operation, the Homalotatic pressure found by 

 the fourth operation, so as to leave an Arrhopic system of externally applied 

 pressures ; whose effects in producing internal stress and strain remain to be 

 found. 



The advantages of the method of decomposing the forces applied to an 

 Elastic Solid arise from the following circumstances : — 



First. It is impossible to determine the effect of any system of forces applied 

 to an elastic solid, unless such system be self-balanced. 



Secondly. It is, if not impossible, extremely difficult to determine directly 

 the effect upon an elastic solid of any self-balanced system of forces which are 

 not all parallel, unless they correspond to an uniform state of stress. 



Thirdly. The difficulties of any problem respecting the stress of an elastic 

 solid are often much increased if the applied pressures are not parallel to an 

 axis of co-ordinates chosen with reference to the figure of the solid. 



It is, therefore, desirable that all those pressures whose effects are not 

 capable of being expressed by a state of stress uniform at every molecule of the 

 solid (like that due to Homalotatic Pressure) should be reduced to a system or 

 systems whose components parallel to any axis whatsoever are self-balanced, 

 and may therefore have their effects separately computed — that is, to an 

 Arrhopic system, or systems ; and this is what is accomplished by the pro- 

 cesses above described. To complete the solution, therefore, of the problem of 

 the internal equilibrium of any elastic solid near the earth's surface, it is only 

 necessary to find the separate effects of three Residual Arrhopic self-balanced 

 systems of parallel pressures, parallel respectively to such axes as the figure of 

 the body may render most convenient. 



(8.) Cases in which the Distribution of Internal Stress is Independent of the 

 Co-efficients of Elasticity of the Solid. 



Theorem. — When the molecular displacements are expressed by algebraical 

 functions of the co-ordinates not exceeding the second degree, and the stresses 

 (consequently) by constants and linear functions of the co-ordinates, the distri- 

 bution of internal stress is independent of the co-efficients of elasticity of the 

 solid. 



Demonstration. — The cases in which the distribution of internal stress is 

 independent of the co-efficients of elasticity, are those in which the number of 

 arbitrary constants in the functions expressing the internal stresses is not greater 

 than the number of arbitrary constants in the functions expressing the mole- 

 cular displacements ; so that, consequently, the internal stresses can be deter- 

 mined from the external forces alone. 



The internal stress at any point is expressed by six components, linear 



