Eoyal Society. £j$ 



their ends at the same temperature and their lateral edges unequally 

 heated. I have no doubt of being able so to verify every thermo- 

 electric characteristic of crystalline structure, in metals in a state of 

 strain. 

 Glasgow College, March 30, 1854. 



P.S. April 19, 1854. — T have today found by experiment that iron 

 wire when stretched by a considerable 'force bears a thermo-electric 

 relation to unstretched iron wire, the opposite of that which I had 

 previously discovered in the case of copper wire ; and I have ascer- 

 tained that when the wire is alternately stretched and unstretched 

 on the two sides of a heated part the current is reversed along with 

 the change of tension, always passing from the unstretched to the 

 stretched part, through the hot locality. 



I hope before the end of the present Session to have a complete 

 account of all the experiments of which the results are stated above, 

 ready to communicate to the Royal Society. 



2. "An Introductory Memoir upon Quan tics." By Arthur Cayley, 

 Esq., F.R.S. 



The subject of Quantics is defined as the entire subject of rational 

 and integral functions, and of the equations and loci to which these 

 give rise, but the memoir relates principally to the functions called 

 quantics ; a quantic being in fact a rational and integral function, 

 homogeneous in regard to a set of facients (j?, y. .), or more gene- 

 rally homogeneous in regard to each of several such sets separately. 

 A quantic of the degrees m, m' .. in the sets {x, y . .) {x\ y'. .) &c. is 

 represented by a notation such as 



(*)(^,y..)V.y--)"'..). 

 where the mark # is considered as indicative of the absolute gene- 

 rality of the quantic. The coefficients of the different terms of the 

 quantic may be either mere numerical multiples of single letters or 

 elements, such as a, ^>, c, or else functions (in general rational and 

 integral functions) of such elements ; this explains the meaning of 

 the expression the elements of a quantic. The theory leads to the 

 discussion of the derivatives called covariants. Of these covariants 

 a very general definition is given as follows, viz. considering the 

 quantic (#) (a;, y..)"^(y,y'.. )""'..), and selecting any two facients of the 

 same set, e. g. the facients x, y, it is remarked tliat there is always 

 an operation upon the elements tantamount as regards the quantic 

 to the operation xdy, viz. if we differentiate with respect to each 

 element, multiply by proper functions of the elements and add, the 

 result will be that obtained by differentiating with dy and multi- 

 plying by X. And if the operation upon the elements tantamount 

 to xd is represented by {xd^}, then writing down the series of ope- 

 rations 



{xdy)— xdy, . .{x'd'y}—x'dy', . .&c., 



where x, y are considered as being successively replaced by every 

 permutation of two different facients of the set {x,y..), x',y' by 

 every permutation of two different facients of the set ix',y' ..) &c., 



