184 
Proceedings of the Royal Society 
A sliglit charge of hydrogen makes the palladium strongly posi- 
tive to the platinum, hut this characteristic gradually diminishes till, 
finally, when the palladium is fully charged with hydrogen, the 
electro-motive force is less than half its original amount. This fact 
seems new, and deserves closer study. 
4. Note on Plane Algebra. By A. Macfarlane, 
M.A., D.Sc. 
By Plane Algebra I mean what Be Morgan called Double 
Algebra, While ordinary algebra deals with quantities which are 
represented on a straight line, and Quaternions with quantities 
which are represented in space. Double Algebra deals with those 
which are represented on a plane. The object of this paper is to 
show some applications of this intermediate method. 
The quantities considered are conveniently denoted by small 
Eoman letters, leaving their Tensor component to be denoted by 
the corresponding Italic letter, and the Versor component by the 
corresponding Greek letter. Thus a denotes a line of length a 
and angle a ; b a line of length &, and angle /?. Quantities of this 
kind are related to those of ordinary algebra as genus and species, 
and the laws of operation for the former are very easily generalised 
from those for the* latter. 
Expansions can be obtained by altering the order of the opera- 
tions performed ; for example, first, by applying the Binomial 
Theorem and then resolving ; and second, by resolving and then 
applying the Binomial Theorem. 
For example — 
a-b a\ a/ a^a2”*'a^'^ 
\ h IP 
— cos ( - ct) 4- ^ COS ifi - 2a) + -3 cos - 3a) -f 
a. Cl (t 
f 1 h IP ) 
+ i\— sill (-a) + -2 sin (/3-2a) sin (2^ - 3a) + | . 
1 ^ 1 
a - b ~ a cos a-b cos p + i [a sin a - 6 sin P) 
Again, 
