1899-1900.] Dr Muir on the Theory of Alternants. 
99 
out at length without specialising n ; but as this does not add to 
clearness or conviction, n may here, for convenience in writing, he 
taken = 4. Let, then, the alternant be 
\a r b s c*d u \ 
so that the multiplier is 
a h _j_ yi _j_ c ^_|_ yh ' 
Expanding the multiplicand first according to powers of «, we 
perform the multiplication by a h ; expanding next according to 
powers of b, we perform the multiplication by b h ; and so on, the 
sum of the products being naturally arrangeable as a square array 
of sixteen terms, viz., 
a r+h \b s c f d u \ - a sJrn \b r c f d u \ + a t+7l \b r c s d u \ - a u+h f r c s d t \ 
- 5 r+7l |a s c^ M | + & s+7l |aVc£ w | - b t+ll \a r c s d u \ + b uJrh \a r dd^ 
+ c r+ll \a s b t d u \ - c s+ll \a r Vd u \ 4- c t+ll \a r b s d u \ - c w+7l |aW*| 
- cT +7l |a s &*c M j + c? s+7l |a r 5 7 c M | - d t+h \a r b s c u \ + d u+h \a r b s c f \ . 
Recombination of these, however, is possible by taking them in 
vertical sets of four, and the result of doing this is 
\d r+h b s c f d a \ - |a s+ W<2“| + \a t+h b r c s d u \ - \a u ^ h b r c s d t \- ) 
so that we have 
||a r 6V<i M | . %a h = |a r+7l &V<i M | + |a r & s+ Vc£ M | + \a r b s c t+li d u \ + |a r &Vc? u+7i |, 
and generally 
| a r b s c f d u e v | . %a ]l = 1 a r + h b s M u e v | + \a r b s+h c f d u e v | 
+ | a r b s c t+h d u e v | + 
The special case where r, s, t, u, ... . proceed by a common 
■difference, h , is drawn attention to, as then all the alternants on the 
right vanish except the last : that is to say, we have 
r r+h r+2h r+(n-l)h ] 
ct n ct n m •••• a .. 
i O' / # * i -y-tii v g TV " **uo l-r illC] 
■a result which may be looked upon as an immediate generalisa- 
tion of one of Cauchy’s. 
When jP>l, the mode of proof is totally different, being an 
attempt at so-called “mathematical induction.” It is not by any 
means readily convincing, and is much less so than it might have 
