124 
Proceedings of Royal Society of Edinburgh . [sess. 
according to descending powers of t 0 , t p . . t m is 
v 
n - m - ] 
M a n - mj 9.n - m+lj • • •) &n) 
n(c> 0 ,ai, . . a n ) 
■effect being given to the sign of summation by permuting in 
every possible way the quantities a 0 , a lf . . a n . 
As has already been seen the expression to be ex, ; nded is equal 
to an aggregate of terms of the form 
^(^0* ^i> • • ■> lm) • 1 t m ) 
f(a)f(b) .... f\p) . (f 0 -a)^- 6) . . . ’ 
where each of the m + 1 quantities a, b, . . ., p is one of the n + 1 
quantities a 0 , a v . . . ‘ Since, however, we are now in search 
of the coefficient of t~ H* 1 . . . we may leave out of account 
all terms of this aggregate which have two or more of the m + 1 
quantities a, b, . . ., p alikej for it has been shown that the ex- 
pansion of such a term cannot contain t-H- 1 • • • t~f° We are 
thus left with an aggregate which may be represented by 
s 
• • •» 
O’"!* • 
tm) 
f { a n-m)f (ct>n-m+ 1 ). • • f ( CL n ) . (t 0 — a n - m )(ti ~ a n -m+ 1 )* • • — ^ n ) 
it being understood that for a n - m , a n - m +i , • • a n is to be taken 
any permutation of m + 1 quantities of the group a 0 , a v . . ., a n . 
But, if the coefficient of t~ l t~ l . . t- 1 in this he denoted by 
H, we have by the first of our auxiliary theorems 
q ff( cl, n-m)Cl>n-m+ 1? • • •? <%n) . U(a n - m ,an-m+ 1? • • • > Q>n) 
^ / {fln-m)f (^bi-m+l) • • • • f (^ w ) 
and using the second to substitute 
(._l)im(m+i)n(a 0> a 1J . . .,a n - m - 1 ) ITL(d Qr a v . . ., a n ) 
for T\-(an - m^n - m+h • * ®n) jf (an-m)f ign-m+l) • • >f (^n) 
we have 
H = • • •’ an - m - 1 ) • $( a n-m ,Cl n-m+\'i « • • ? & n ) ? 
where, be it remembered, the n+ 1 elements a Q , oq, ; . ., a n are 
to be separated in every possible way into two classes containing 
