Mr Babbage m a Theorem in Prime Numbers. 49 
which we have found in those deducted from the co-efficients of 
the square of a binomial. 
In case 1 should be a prime number, we may find for- 
mulae divisible by (?^- — 1)®> thus : If instead of subtracting the 
two extreme terms of tlie co-efficients, we subtract the two 
first and two last, we have. 
^ f w-f 1 
^ \ 1 . 2 .. . 71^1 
and since, if n — 1 is a prime, each term on the right side of the 
equation is divisible by (n — 1)*^ ; the expression on the left side 
is always divisible by (n — 1 )^ in the same circumstances; or 
the expression 
fi-l-l.n-^2...2n — 1 ^ 
2 ^ 
1.2.. ..n — 1 
is always divisible by (n — 1)^, if 1 is a prime number, 
otherwise it is not divisible. It is obvious that similar theorems 
might be deduced, in which the prime divisors should be 
{n — 2Y, or {n — 3)^, &c. 
Art. VIII. — Description of the Diamond Mine of Panna. 
By Francis Hamilton, M. D. F. R. S. & F. A. S. Lond. Sr 
Edin. Communicated by the Author, 
During the rainy season of the year 1813, on my way 
from Agra to Chunar, I made an excursion from the Yamuna 
{Jumna., Rennell,) to visit the Diamond Mine at Panna, and 
first proceeded up the Ken in my boats for two days ; but I 
made very little progress, owing to the strength of the current, 
and the badness of the ground on the bank for tracking. The 
Ken is not a great deal smaller than the Yamuna, and resembles 
it much in having very high banks intersected by numerous 
ravines. Its channel abounds in pebbles of agate and jasper; 
but, in the rainy season being entirely filled with water, scarcely 
any were procurable; nor did I obtain any good specimens. 
These pebbles are not so much variegated by zones of different 
VQL. I. NO. I. JUNE 181 9> M 
