720 PROFESSOR MACQUORN RANKINE ON THE DECOMPOSITION OF 



(6.) Decomposition of Abarytic Pressures. 



Definition. — Homalotatic System of Pressures : A system of Abarytic Pres- 

 sures applied to the surface of a body, and producing an uniform state of stress 

 throughout its internal molecules. 



Prop. IV. — Problem. To Decompose any Abarytic System of Pressures 

 applied to the Surface of a Solid into a Homalotatic System, and an Arrhopic 

 System. 



Having computed the six Rhopimetric co-efficients for the given Abarytic 

 system of Pressures, as referred to any set of orthogonal axes, take the follow- 

 ing values for a set of six uniform internal stresses ; V being (as in Proposi- 

 tion III.) the volume of the solid : — 



N - A • N -^~- N - ° 



±y x-0 — -it > - L> y • — ^Tr~ 5 ■** z • — 



y > —y o y , — z-u — y , 



y j J-yo — y j J-z-o — y , 



(8). 



These quantities being constant, fulfil equation (7). 



The corresponding external pressures are as follows, according to equa- 

 tion (4) : — 



P = y- { n x A + n y ~F + n z E I 



Q = 4 { »JF + n y B + n z T> j j> (»)• 



R ° = "v j ^ E + n P + n & \ 



If the Homalotatic system of pressures given by these equations be taken from 

 the entire Abarytic system, an Arrhopic system will remain. 



For the Rhopimetric co-efficients corresponding to the Homalotatic pres- 

 sures are as follows : — 



A =ffx'P d 2 s 



= y- \ Affx ' dydz + Yffx'dzdx + Effx'dxdy \ 



(and similar equations for the others, mutatis mutandis). 

 Now, observing as before, that 



dydz, dzdx, dxdy 

 are to be considered positive or negative according to the sign of 



^x> My> f^z} 



it appears that 



