202 



tain processes of derivation, distributive, but not necessarily commu- 

 tative. 



Each primitive function must satisfy the condition 



/ rf 2 d* d* \ , 

 [ j __ _i_ )$=0, 

 + 



and may belong to one or other of eight classes, according as it is 

 even or odd with respect to x, y and z. 



The processes of derivation applicable to the primitive functions 

 contain three arbitrary constants for each primitive function. Hence 

 when there is a series of primitive functions of different orders, there 

 are twenty -four arbitrary constants for each order of terms. 



In the developments of the residual external pressures, there are 

 also twenty-four constant coefficients for each order of terms, of 

 which the arbitrary constants are linear functions. 



The notation of M. Lame's work on the Mathematical Theory of 

 the Elasticity of Solid Bodies, so far as it relates to isotropic sub- 

 stances, is compared with that of this paper. 



Reference being made to the known method of representing the 

 elastic pressures at a given particle of a solid, in magnitude and di- 

 rection, by the radii of an ellipsoid, and the positions of the surfaces 

 to which those pressures are applied by those of the tangent planes 

 to an ellipsoid or hyperboloid, the difference (not generally attended 

 to) between the cone of tangential pressures, and the cone of sliding, 

 is pointed out. This . difference is important in the theory of the 

 strength of materials. 



