272 
MR. MACQUORN RANKINE ON AXES OF 
We may regard the Thlipsinomic Coefficients, like the Tasinomic Coefficients, as 
binary compounds of the following six Umbrae, 
(a), (b), (c), (/), (m), (n), 
which being respectively substituted for 
P„ P^, P 3 , 2 Q„ 2Q„ 2 Q 3 
in the equation of the Tasimetric Surface (4.), produce the following equation of the 
Umbral Thlipsinomic Ellipsoid, 
{m)zx-^{nt)xy=\, (28.) 
from which, by involution, multiplication, and other operations exactly analogous to 
those performed on the Umbral Tasinomic Ellipsoid, there may be deduced the 
equations of Thlipsinomic Surfaces exactly corresponding to the Tasinomic Surfaces 
already described; while, from the Umbral Matrix, 
(«) i(m) 
m (b) \{l) 
(29.) 
l{m) !(/) (c) J 
may be formed Thlipsinomic Invariants corresponding to the Tasinomic Invariants. 
Hence it appears, that every function of the Tasinomic Coefficients is converted into 
a function of the Thlipsinomic Coefficients with analogous properties, by the substi- 
tution of Thlipsinomic for Tasinomic Umbrae according to the following table : — 
Tasinomic Umbrae (a), (j3), (y), (X), {(ju), (v), 
Thlipsinomic Umbrae .... (a), (b), (c), \{l), \{m), ^(n). 
Amongst the Thlipsinomic Invariants may be distinguished the Cubic Compressibility, 
which is formed by squaring the umbral invariant (a)-|-( 6 )-|-(c), and has the follow- 
ing value : 
(a") -f ( 6-) -1- (c") -h 2 ( 6c) -b 2 (ca) -f- 2 (a &) . 
15. Thlipsinomic and Tasinomic Contragredient Systems. 
Let the following square matrices be formed with the Tasinomic and Thlipsinomic 
Coefficients respectively : — 
(a[3) 
(ya) 
{aX) 
(av) 
(a^) 
(ab) 
(ca) 
(al) 
(am) 
(an) 
(«^) 
m 
CM 
{(5X) 
((Bv) 
{ah) 
(be) 
(bl) 
(bm) 
(bn) 
(y«) 
(M 
(f) 
(yX) 
ivy') 
(yv) 
(ca) 
(be) 
(cl) 
(cm) 
(cn) 
(aX) 
m 
(yX) 
(¥) 
{vX) 
(al) 
(bl) 
(cl) 
cn 
(Im) 
(nl) 
(a/x) 
Cry^) 
(/^^) 
(y.u) 
(am) 
(bm) 
(cm) 
(Im) 
(m^) 
(mn) 
{yi>) 
((XV) 
f) 
(an) 
(bn) 
(cn) 
(nl) 
(mn) 
Then will these matrices be mutually the two systems of coefficients arrayed 
in them, with their respective systems of functions, mutually contragredient, and 
