MATHEMATICS: O. E. GLENN 
147 
This infinitude of concomitants therefore forms a system which possesses 
the property of finiteness, 4 and the fundamental invariants are furnished by 
the finite set of irreducible solutions of 5 = 0. 
In the ternary realm the lines joining the three poles of the transformation 
T on three variables furnish three linear forms in terms of which any quantic 
f m in three variables can be expanded. The coefficients in this expansion 
urnish complete systems in each of several domains. In particular, if T 
is the transformation which rotates cartesian axes in three dimensional space, 
x = h%' + hy' + hz', 
y = m\x' + why' + W3Z 7 , 
2 = n\x' + n^y' + 
the coefficients being the well-known direction cosines of three axes, the 
invariant triangle on the poles consists of the lines 
Ui m (h + n x e ±id ) x + (m 8 + n 2 e ±iQ ) y + (». - h + l^e^ 9 + e* 2ie ) z = 0, 
fo = (h + ni) x + (m 3 + fh) y + (»3 — /i — M2 + 1) z = 0, 
where 0 is a definite auxiliary angle. The coefficients in the expansion of f m 
in the arguments f ±h f 0 are invariants belonging to the domain of complex 
numbers, while the finiteness of complete systems in the real domain is de- 
termined, and the fundamental concomitants are given, by the finite set of 
irreducible solutions of the linear diophantine equation 
m m-i 
2 2 xf (m-i-2j)+(3-a= 0. (4) 
i=0j = 0 
The Invariants of Relativity.— Among numerous important particular cases 
of the above theory is the transformation of space and time coordinates in the 
theory of relativity, known as the transformation of Einstein. 5 This consists 
of 
Ti: t = n(cH' + vx')/c, x = n {vt' + *') c, y = y', z=z\ 
where /i = V( 6 ' 2 ~~ " ^ 2 ) is the time, c the velocity of light, v the relative 
velocity of the moving systems of reference and x, y, z space coordinates. 
For these unitary substitutions, £ = ct + x, rj = ct — x and the invariant 
relations are 
r = pi, v' = p-% 
in which (Cf. (2)) 
P = a/ c — v / y/ c + v . 
We now find 
J : c 2 * 2 - x\ 
