920 Proceedings of Royal Society of Edinburgh. [sess. 
and the proof, it is stated, consists in taking the equations 
h = adJ + yY + zW, U = £u + , 
h x = xfJ + y x Y + iqW, V = rju + v) Y v + rj 2 w , 
h 2 = x 2 V + y 2 V + z 2 W, W = lu + t\ v + ^ 2 W > 
deducing from them two different triads of equations independent 
of U , V , W , solving these triads for u , v , w , and then comparing 
the results.* 
In connection with this two points have to be noted. The 
first is the use of the notation 
H I, 
which may be compared with Catalan’s of the year 1846, and 
with the modification of the vertical-line notation which the 
printers of Crelle’s and Liouville’s journals employed for two or 
three years in setting up Cayley’s papers. That Joachimsthal was 
familiar with Catalan’s paper of 1846 is made more probable by 
the occurrence of a footnote (p. 28) giving 
x +1 y z 
x y z 1 
1 | 
\1 V * ' 
- x + V y' z 
- — det.- 
.r y' z 
[ + det.j 
r v * - 
y" z" 
x' y" z" J 
! 1 
\ i" y" z' 
an identity which Catalan was the first to formulate in a similar 
way. 
The second point is the unnaturalness, in view of the mode of 
proof, of not writing the second determinant of the theorem in 
the form 
ft 
Vi 
ft 
* The details of the proof not being given, one cannot guess how it was 
that a second theorem was not obtained, viz. , the theorem 
h 
+VVi +*(i 
+w 2 +«C 2 
1 1 1 
h 
I 2 I 
K 
^ili + Vivi + ^iCi 
Xlk + VlVi + tlCz 
= j | xfi 1 z 2 | 
Vi 
V2 j 
K 
+ 2 / 2^1 + z zC\ 
X 2%2 + 2/2^2 + ^2^2 
1 1 xyM | 
Ci 
C 2 | 
