FORCES EXTERNALLY APPLIED TO AN ELASTIC SOLID. 723 



functions of the first differential co-efficients of the molecular displacements, 

 which last are three in number. 



When these are expressed by transcendental or other irrational functions, 

 the former number of constants is double of the latter ; hence this class of 

 cases is excluded. 



When the molecular displacements are expressed by three homogeneous 



rational and integral functions of the three co-ordinates of the n th degree, the 



3 

 number of arbitrary constants contained in them is r> (n + 1) (n + 2). 



In the same case, the component stresses are expressed by six rational and 

 integral homogeneous functions of the degree n — 1 ; so that in them the 



number of arbitrary constants is -^n (n + 1). 



Hence the ratio borne by the number of arbitrary constants in the expres- 

 sions for stresses to the corresponding number in the expressions for molecular 

 displacements, is 



2n 

 n + 2 ' 



This ratio is less than, equal to, or greater than unity, according as n is less 

 than, equal to, or greater than 2 ; therefore the distribution of stress is inde- 

 pendent of the co-efficients of elasticity for molecular displacements expressed 

 by rational functions not exceeding the second degree, and stresses expressed 

 by constants and by linear functions of the co-ordinates. Q . E . D. 



As n increases indefinitely, the above-mentioned ratio approximates to 2, 

 being its value for irrational functions. 



(9.) The Classes of External Pressures which produce stresses answering the 

 preceding description are the following : — 



Homalotatic Pressures, for which the stresses are expressed by constants, 

 and Antibarytic, Homalocamptic, and Homalostrephic Pressures, for which the 

 stresses are expressed by linear functions. 



It has already been seen, that for the systems of pressures designated as 

 Homalotatic and Antibarytic, the internal stresses are determined independently 

 of the co-efficients of elasticity of the body, being in the former case uniform, 

 and in the latter consisting in a vertical normal stress, which is a linear function 

 of the vertical ordinate from the horizontal plane of the body's centre of gravity. 

 The consideration of these two systems of stresses forms part of the solution of 

 every problem concerning the equilibrium of an elastic solid. 



Definitions. — Homalocamptic Pressures (or pressures of Uniform Bend- 

 ing). — A system of external pressures corresponding to a system of normal in- 

 ternal stresses uniform in direction, whose intensity is a linear, function of an 



VOL. XXVI. PART IV. 9 B 



