1899—1900.] Dr Muir on the Theory of Alternants. 
127 
according to descending powers of t 0 , t 15 . . ., t m , inhere 
f( x ) = (x - a o)( x - a i) • • • (x-a„). 
This is followed up by actually working out the expansion in 
question, the numerator being of course changed into 
and its cofactor 
into 
2±CC 
1 1 
/(* 0 ) * M) 
t 
( -i > 
C 1 C 2 
+2 T 77+3 ^ 
0 0 
( L + A 
\ ,77 + 1 ^ ,77+2 
vr o r o 
/_1 + Cl + A + 
y^77+l T jn+ 2 T ^n+S ^ 
+ 
1 
f(tm) 
C s 
/II -f- 1 5 
+ -^- + 
7 ^w+i+* 
o 
Ci ^ c, 
j ±_ _] + 
71+1 T ^77+2 T ^77+3 T 
c s 
where C s is the sum of all the products of s elements, different or 
equal, taken from a 0 , a v . . ., a n . Multiplication of these m + 1 
partial factors has next to be performed, the general term of the 
result being seen to be 
Cg () C.9] . . . c$ m 
7l+l + S(),7i + l + Si 77+1 + Swi 
h U * * * * m 
All that remains, then, is the multiplication of this result by the 
corresponding expression for the original numerator, i.e ., by 
± t™t™ 1 . . . t m -i, which, be it noted, consists of (m+ 1) 2 
terms, the % referring to permutation of the indices m, m — 1, . . ., 
1, 0. Without further delay, Jacobi merely adds that the general 
term will therefore become 
2 ± 
CUA, g g | 
ti-7?i+1+s 0 n-m+l+si 
*0 \ 
w+l + Sm 5 
and that consequently the proposition last formulated will 
{ suggest ’ the identity 
