On the Distribution of Electricity on Metallic Bodies. 269 
expression for the eoth root of the matrix: and we see that the 
number of such roots is &)', where i is the order of the matrix. 
When, however, the matrix is unitary, i. e. all its terms except 
the diagonal ones are zeros or zeroidal. i. e. when all its terms 
are Zi rosj this conclusion is no longer applicable, and a certain 
definite number of arbitrary quantities enter into the general 
expressions for the roots. 
The case of the extraction of any root of a unitary matrix 
of the second order was first considered and successfully 
treated by the late Mr. JBabbage ; it reappears in M. Serret's 
Cours d'Algebre supe'rieure." This problem is of course 
the same as that of finding a function 7—7—^ of any given 
order of periodicity. My memoir will give the solution of the 
corresponding problem for a matrix of any order. Of the 
many unexpected results which I have obtained by my new 
method, not the least striking is the rapprochement which it 
establishes between the theory of Matrices and that of Inva- 
riants. The theory of invariance relative to associated Matrices 
includes and transcends that relative to algebraical functions. 
XL. On the Influence of the Direction of the Lines of Force 
on the Distribution of Electricity on Metallic Bodies. By 
Alfeed Tbibe, F.Inst. C, Lecturer on Chemistry in Dul- 
icich College*. 
ITTISH it to be understood at the outset that the results 
on electrical distribution, to which reference will be made 
in this paper, were obtained during the electrolysis of a solu- 
tion of copper sulphate, by determining the amount, extent, 
position, or nature of the electrochemical action set up on a 
metallic plate, or other-shaped conductor, immersed in the 
electrolyte, but not in metallic connexion with the battery- 
electrodes. I would also point out that when I speak of the 
lines of force as having a certain direction with regard to a 
part or parts of a metallic plate, or other-shaped analyzer, or 
as having a certain direction with regard to the boundary of 
an electrochemical deposit, it is to be understood that such 
would be the direction supposing the analyzer itself exercised 
no disturbing influence on these imacrinarv lines in the field 
01 action. 
I. When the direction of the lines tj ft - is parallel to the 
sides and perpendicular to the ends of the metallic conductor. 
Some six years ago (Proe. Roy. Soc. 1877, no. 181), I 
* Communicated bv the Author. 
