122 Dr. Brinkley’s Investigation of the general Term 
(±)- n + l 4. 
b n - 1 \ * I 
+ * (t) h + 
P (t) ^ + &C. 
Whence the truth of the theorem is manifest. 
Theorem V. S" u — if) f\ n ux n - ] — fl.” 1 ux n 1 4- . . &c. 
\h n l 1 h n—i 1 
where the numerical coefficient of the m + 1 term is the co- 
efficient of h m in the expansion of ( 1 -f- ~ -f- ~~ 3 + &c.)“ K . 
The demonstration of this is contained in that of the last 
theorem. 
Previously to the investigation of the numerical coefficient 
of the general m-j-i term, it will be convenient to premise 
the following lemmas. 
Lemma I. Let n represent any affirmative integral num- 
ber, and m any other affirmative integral number not greater 
than n. Then 
j'iy p’j* | i ? Y [ i''] w &c. = A m o n where p, q, r, &c. : t, v, w, &c. 
represent any affirmative integral numbers satisfying the 
equations &c. = m 
ip f-vqf-zur— {- &c. — n. 
Demonstration . 
T . (0 n 
Let o' = 1 
= 1.2 i 1 ^)^ w ^ ere t-\-v—<2. & tp-\-vq=n. 
a [3) = 1.2.3^ 1 ?) 1 (l*)* (r) w where t- {.v-\-w=$ and tp-f 
vqfrw—n. 
flf) ( i^(tj w &c. where t+v+w+&c. 
==m and tp-\-vq-\-rw- f- &c. —n. 
