922 
Proceedings of Royal Society of Edinburgh. [sess. 
From these last equations, on multiplying by 
0A 0A. 0A 
d$n ’ fyn ’ ’ 9r« 
respectively, and adding, there is obtained with the help of the 
known identities 
0 A 
0 A 
BA 
= 0 , 
dL 
' ' +t °bT 
n 
0 A 
0 A 
0 A 
= 0 , 
ZL 
+ Vi + • • 
dy n 
• • + T 1 — 
dr v 
0 A 
0 A 
0 A 
% 
+ T 
the result 
0D 0A 01) 0 A 
dx^dL + dy^d Vn + 
0D0A 
dL dr„ 
= A, 
m 
dL 
z.e. 
det.- 
+ det. 
' Vo 
^0 
' Vo 
Co 
• T 0 
* 
*1 • 
- . det.- 
V\ 
ii 
• T 1 
< Vn-i 
1 • 
• . 
. 
. Vn- 
-1 L- 
-1 • • 
• T n-1 
r X 0 
Zq 
K 
> 
^0 
Co 
• T 0 
X 1 
h • 
h 
1 . det.- 
tl 
Cl 
* T 1 
v X n - 1 
*-l • 
L — 1 
) 
. fn- 
-1 in- 
-1 • ’ 
• T n- 1 
1 
( k 
,0 
lo 
,i 
1 0 , ti- 
| 
= 
det.- 
1 K 
,0 
h 
, i 
ll ,n- 
-i 1 
1 
Jn- 
-1,0 
In 
- 1,1 • * 
K- 1, 
,n - 1 j 
+ 
Using later phraseology we may say in describing this that from 
the theorem regarding the product of two square matrices of 
order ra+1 there is obtained the theorem regarding the product 
of two rectangular but non-quadrate matrices, the latter product 
appearing as a principal coaxial minor of the former. Cauchy’s 
generalisation concerned any minor of the former product, but 
even this further extension was not beyond Joachimsthal’s reach, 
for he ends with the remark “ En differential de nouveau par 
rapport a x n _ x , y n _ x , ... on obtiendra d’autres formules ; et ainsi 
de suite.” 
