532 Proceedings of Royal Society of Edinhurgh. 
1 1 «2 «3 «4 \ I «1 «5 \ 
1a,A2A3)-|a,aJ 
is fixed by making it the same as the sign of the term 
0,2 <*3 0(4 dl (Zg 
Aj A2 Ag A^ Ag ; ' 
but nothing is said as to how this ensures that the 11 other terms 
of the product shall have their proper sign. 
Considerably less interest attaches to the next theorem dealt 
with, — Vandermonde’s theorem regarding the effect of the equality 
of two upper or two lower elements. All that is fresh is the 
lengthy demonstration by the method of so-called induction. The 
identities immediately following from it by expansion Schweins 
expresses as follows : — 
^q-iaq(^q+l 
• • -^*-1 
• “«+] 
■^5 — 1 Eq • 
where x = I , 2 , . . . . 
• • • aq _ a 
• A„ 
, n. 
(xii. 10.) 
This concludes the first chapter of the first section. 
The second chapter deals with a most notable generalisation, and 
is worthy of being reproduced with little or no abridgment. The 
subject may be described as the transformation of an aggregate of 
products of pairs of determinants into another aggregate of similar 
kind. A special example of the transformation is taken to open 
the chapter with, the initial aggregate of products being in this case 
+ - |aiVs5'4l-l<^6«6/?l 
Expanding the first factor of each product Schweins obtains 
I — d^a-jh^c^ + d^af^c^ — .\e^fQgrj\ 
— I ~ ^31^1^2^41 d* ~ ^ll^2^3^4l } 
+ { /4l^l^2^3l “ /3i^l^2^4l ~~ /ll^2^3^4l } 
- - g^\af^c^\ g^\af.f^ - gi\af^c^}.\d^ej^\. 
