338 
Proceedings of the Royal Society 
2. Specimen of Auriferous Quartz. By Patrick Dudgeon. 
The specimen was found near Wanlockhead, Dumfriesshire, in 
1872, by Andrew Gfemmell, miner, who unfortunately broke up the 
piece, and disposed of the fragments to different parties in the 
locality. Mr Dudgeon was enabled to get possession of all the 
fragments, and has restored the mass to its original form. The 
specimen is a very interesting one, being the largest known piece 
of auriferous quartz found in Scotland ; and Mr Dudgeon thought 
it would interest the fellows of the Society to exhibit it at the 
meeting before placing it in the Museum of Science and Art, which 
has now been done with the consent of the different owners. 
The following Gentlemen were elected Fellows of the 
Society : — 
George Broadrick, C.E., Claremont Cottage, Leith. 
John Napier, Engineer, Lancefield House, Glasgow. 
James King, of Campsie, 12 Claremont Terrace, Glasgow. 
George Cunningham, C.E., 2 Ainslie Place. 
Monday , Vdth March 1877. 
Sir WILLIAM THOMSON, President, in the Chair. 
The following Communications were read : — 
1. On a Problem of Arrangements. By Professor Cayley. 
It is a well-known problem to find for n letters the number of 
the arrangements in which no letter occupies its original place; 
and the solution of it is given by the following general theorem : — 
viz., the number of the arrangements which satisfy any r condi- 
tions is 
(1 - 1)(1 - 2) .... (1 - 0 , 
= 1-2(1) + 2(12) . . . ±(12 . . .r), 
where 1 denotes the whole number of arrangements; (1) the 
number of them which fail in regard to the first condition ; (2) 
the number which fail in regard to the second condition ; (12) the 
number which fail in regard to the first condition, and also in 
