103 
it and the negative carbon was about 15 volts and between it and the posi- 
tive carbon about 40 volts. The introduction of Ca or K into the negative 
earbon did not change the voltage between it and the third carbon. When 
the salt was introduced into the positive pole the voltage between the 
positive pole and the third carbon fell to 25 volts, but the voltage between 
the negative and third carbons remained 15. It appears that the current 
passes from pole to pole, in part, at least, as a convective discharge of 
charged particles. 
Note oN CHARLES SmItTH’s DEFINITION OF MULTIPLICATION. By ROBERT 
J. ALLEY. 
‘*To multiply one number by a second is to do to the first what is done to 
unity to obtain the second.” 
This definition covers the multiplication of positive and negative integers, 
fractions and imaginary numbers. Accepting it as true, the law of signs follows 
as a result. We can easily show that it includes the multiplication of imaginaries. 
Suppose we are to multiply a by bi. We are to do to a what is done to unity to 
obtain 6%. To obtain bi from unity we take unity 6 times and turn it counter 
.clock-wise through an angle of 90 degrees. By performing this operation upon 
a we obtain abi. Suppose we are to multiply ai by bi. By the same process as 
above we readily see that the result is — ab. This shows that the definition 
includes practically all of Quaternion multiplication. 
If we undertake to apply it to the multiplication of 6 by a? we encounter our 
first difficulty. a? has been obtained from unity by taking unity a times and 
squaring. If we do this to b we obtain a* b’, a result manifestly wrong. Ii, 
however, we remember that a=aa and is obtained from unity by taking it aa 
times, our difficulty disappears and we obtain the correct result a2 b. If we under- 
take to multiply b by a% we find a difficulty which seems to be insurmountable. 
The only way we can obtain a” from unity is by taking unity a times and extract- 
ing the square root. If we do this to b we obtain the incorrect product a b”. 
The definition seems to fail utterly when applied to irrationals. Perhaps, after 
all, it is better to follow the custom of most algebras and make only symbolic 
definitions. 
Inprana University, December 8, 1897. 
