FORCES EXTERNALLY APPLIED TO AN ELASTIC SOLID. 721 



ffx'dydz = V ; ffx'dzdx = ; ffx'dydz == ; 

 so that 



A = A ; 



and in like manner, each of the Rhopimetric co-efficients for the system of 

 Homalotatic Pressures is proved to be equal to the corresponding co-efficient 

 for the entire system of Abarytic Pressures ; so that the system of residual 

 pressures, left after taking away the Homalotatic Pressure from the Abarytic 

 Pressures, is Arrhopic. Q. E . F. 



Remark.— The Homalotatic system of six uniform stresses obviously repre- 

 sents the mean stale of st?*ess of the whole body. 



Corollary. — The set of six uniform stresses given by equation (8) are 

 equivalent to three principal rectangular uniform normal stresses along the Isor- 

 rhopic Axes. For the three principal normal stresses of the system (8) are in 

 direction parallel to, and in magnitude represented by, the reciprocals of the 

 squares of the principal semi-axes of the surface, 



N,. a>"+-N f . y« + lC *" 



+ 2T x . yz + 2T y . zx + 2T z . xy=l; . . (10), 



which is similar and parallel to the Rhopimetric surface. 



(7.) Recapitulation, and Statement of the Advantages of the Method described. 

 — The following is a summary of the processes of the before-described method 

 of decomposing any self-balanced system of forces applied to an Elastic Solid 

 near the earth's surface : — 



First. By Proposition II. Equation 2, find the six Rhopimetric Co-efficients 

 of the system of applied forces, including gravitation. 



Secondly. By Proposition III. Equations 5, 6, compute the system of verti- 

 cal Antibarytic Pressures, with their corresponding vertical internal stresses ; 

 which pressures at once balance the force of gravitation, and form with it an 

 Arrhopic system. 



Thirdly. Take away from the entire pressure applied at each element of the 

 body's surface the Antibarytic pressure at the same element, so as to leave a 

 system of Abarytic Pressures, which is self-balanced, independently of gravi- 

 tation, and whose Rhopimetric co-efficients are the same with those originally 

 computed. 



Fourthly. By Proposition IV. Equations 8 and 9, compute from the given 

 set of Rhopimetric co-efficients the system of six uniform mean internal stresses, 

 with the corresponding system of Homalotatic external pressures. 



Fifthly. Take away from the Abarytic pressure on each element of the 



