264 
MR. MACQUORN RANKINE ON AXES OF 
and the strains 
P., P„ P3, 2Q., 2 Q„ 2Q3 
iimst be respectively covariant with the squares and products, 
y"^, z^, 2yz, 2zx, 2xy. 
3. Thlipsimetric and Tasimetric Surfaces and Invay'iants. 
Isotropic functions of the elementary strains and stresses, which may be called 
respectively Thlipsimetric and Tasimetric Invariants, are easily deduced from the 
principle, that the strains may be represented by the coefficients of the following 
Thlipsimetric Surface, 
ax'^-\-^y^-{-yz’^-{-'kyz-\-y!jZX-\-vxy=.\, (3.) 
and the stresses by the coefficients of the Tasimetric Surface, 
^f-{-V^y‘^-\-V^z‘^-{-2Q.,yz-\-2Q.,zx-\-2Q.^xy=\ (4.) 
These surfaces, and others deduced from them, have been fully discussed by 
M. Cauchy and M. Lame. 
The invariants in question may all be deduced from the following pair of contra- 
gredient matrices ; — 
For Strains. 
“22 
For Stresses. 
P, Qs 
2 
Qs P 2 Qi V (5 A.) 
Qs Q. Ps 
The following are the primitive thlipsimetric invariants, from which an indefinite 
number of others may be deduced by involution, multiplication, addition, and sub- 
traction : — 
a+/3+y=^i (the cubic dilatation) ; 
-f 705 + ap — i H- -f y") = ^2 ; 
— i = ^s- 
( 6 .) 
The Potential Energy U is what Mr. Sylvester calls a “ Universal Mixed Con- 
comitant,” its value being 
U= — KPia+Ps/S-l- Pay-j-Qi^t-J-Qsi^+Qs*') (7-) 
4. Tasinomic Functions, Surfaces, and Umhrce. 
If, in any isotropic function of the coordinates and the elementary strains, there 
be substituted for each square or product of elementary strains, that Tasinomic 
Coefficient which is covariant with it, the result will be an Isotropic Function of the 
Coordinates and Tasinomic Coefficients, called a Tasinomic Function. 
