112 
MATHEMATICS: L. E. DICKSON 
Proc. N. A. S. 
has the term 2x2^C2-Xz^c 3 which does not occur in <r(#)<r(£). But c 2 c 3 4:0 
by hypothesis. Hence n> 3. 
Hitherto we have not examined the conditions which follow from the 
final equations (3) ; these are 
7jik = -Jijk (*", i = 2, . . ., n\ i 4= /). (10) 
6. Taking n = 4 and applying (10), we see that (9) become 
{ Xi = x£i + C2X2Z2 + + c&Jfa X 2 =xi£ 2 + «2fi + 7342(^4-^3), /.^ 
) X 3 = *l£s + + 7243 (#2& — #4&) , X 4 = + X&l + 7234 (*2& - *3&) • 
These transformations do not in general form a group and hence are not 
generated by the corresponding infinitesimal transformations employed 
above. Hence it remains to require that <r(X) = <r(x)<r(g) under the 
transformations (11). The conditions are seen to be 
2 2 2 
C3C4 = — C27 342, ^4 — ~ C37 243, ^3 = - Q7 234, ^47234 == ^27342 = — ^37243, 
the first two of which reduce to the third by means of the last three equa- 
tions. To these last can be reduced all the conditions (8) by means of (10). 
Applying the transformation of variables which multiplies x it £ 4 , X A 
by 7234, and leaves the remaining x t , X, unaltered, we get 
/ Xi = x£i + C2X2& + — C2P2X&, X 2 = xi£ 2 + X2Z1 - c 3 x 3 %4 + cax&, q 1 ^ 
\X 3 = #l£ 3 + %£l + C2X2U — C2X A %2, ^4 = ^4 + ^1 + ^3 — ^2. 
These are the values obtained by Lagrange 5 in his generalization <r(#)<r(£) = 
<t(X) of Buler's formula for the product of two sums of four squares. 
Then = X gives the following multiplication table for the units : 
(e\ = c 2 , el = c 3 , e; = — c 2 c 3 , e 2 e 3 = e 4t e 3 e 2 = —e if 
\ 02^4 = £2^3, ^2 = — £2^3, ^3^4 = — ^2, £4#3 = C 3 e 2 . 
This algebra is associative and is the direct generalization of quaternions 
to a general field F which the writer 6 obtained elsewhere from assump- 
tions including associativity. The four-rowed determinants of the general 
number x of this algebra equals 0 2 (x). The case c 2 = c 3 = —1 gives the 
algebra of quaternions, for which it is customary to write i, j, k instead 
of our units e 2 , e 3 , e*. 
7. It is not very laborious to show by the above method that the cases 
n = 5 and n = 6 are excluded. However, Hurwitz has proved that a 
relation of the form v(x)<t(£) = <j(X) is impossible if n 4 1 1, 2, 4, 8. A 
slight simplification of his proof, together with an account of the history 
of this problem, has been given by the writer. 7 Hurwitz made no attempt 
to find all solutions when n = 4. We proceed to treat this problem. 
Consider the case 04 = —1 to which the general case may be reduced 
by an irrational transformation. Then <r(x) = 2#*. We investigate 
the linear algebras having property (4), i. e., 
(*:+... + x 'm: +... +q = x\ + . . . + xi (13) 
