12 Prof. Wartmann's Researches on the 



algebra, and has given the rule for the like arrangement of 

 12n + 3 symbols, whenever 6w + l is a prime number. It thus 

 appears, that all the results which are indicated by the analysis 

 ot Mr. Spottiswoode in this Journal of May last, and which were 

 anticipated, rather than affirmed by him, are certainly true. 



The arguments adduced by me (page 526, vol. iii.) to show 

 that a similar arrangement of 21 and of 33 symbols is impos- 

 sible, while they are, as I conceive, conclusive for those numbers, 

 are not tenable for the case of 3w, when n is any prime greater 

 than 11 ; for the reason that the triads, required for such arrange- 

 ment, are fewer than those which can be made with n things. 

 I would beg leave to add the following to the theorems which I 

 have already had your permission to enunciate. 



Ifv is a prime number f (r** — l):(r— 1) symbols can be arranged 

 m (v -{-Vj-plets till every duad is once and once only employed. 



IfVy (r*— l):(r — 1), (r^ — l):(r— 1), S^c. he prime numbers, 



(r + 1). (-^—y j • ( y) ••••( T-) symbols can be ar- 

 ranged in {Y-hl)-plets till every duad has been once and once only 

 employed, whatever be the number and the powers of the prime 

 factors. 



Croft, near Warrington, 

 December 3, 1852. 



V. Researches on the Conductibility of Minerals for Voltaic 

 Electricity. By Elie Wartmann, Professor of Natural Phi- 

 losophy in the Academy of Geneva^. 



IN examining the conductibility of crystals, the author, as his 

 title indicates, has made use of voltaic electricity, the effects 

 produced being measured by a good galvanometer introduced 

 into the circuit. If the mineral appeared to be an insulator, the 

 intensity of the current was augmented until a battery consisting 

 of ten large couples of Daniell was applied. In the contraiy 

 case the current was moderated to the required degree by means 

 of a rheostat. 



The purity of the minerals exercises a great influence on their 

 conductibility; the author therefore employed, in all possible 

 cases, well-determined crystals. All the experiments were made 

 in winter, in a room whose temperature varied from 14° to 17° 

 Cent., between which Hmits the conductibility of the substance is 

 not sensibly altered. Every experiment was repeated several 



• Abstract of a paper read before the Society of Physics and Natural 

 History of Geneva. 



