288 Dr. A. Macfarlane on the 
By the ordinary process of solution of simultaneous equa- 
tions we get 
: _ cetbf 
cd—af 
= ae+bd 
ae+bd 
and y = 
Now, according to Boole’s method, 
- - _ = A,abedef + A,abedef’ + A;abcde/f + A,abedelf’ 
+ ~ + + 
+ A,a’b'c'd/ef + Agsa’b'c'd’ ef” + Agza'b'c'd' elf 
+Agsa’b’e'd'ef’; 
where the coefficients A,, A,,...Ag, are numerical. The 
coefficient for any term is found by supposing that term 
coextensive with the universe, and substituting 1 or 0, as the 
ce + bf 
RE If U abcdef 
is identical with the universe, then a, 0, c, d, e, fare each 1, 
and : 
case may be, for each of the letters in 
goggle sl 
AG ee 
If U abcdef’ = U, then f is 0 and all the others 1, and 
A.= 
According to Boole, these coefficients are susceptible of one 
or other of four interpretations: 1 means all, 0 means none, 
ble 
9 Means none or a portion or all, and every other coefficient 
shows that the term is impossible. 
When I sought to determine the coefficients for # and y by 
the above method, I found that many coefficients assumed the 
indeterminate form ; and I found, on verifying the solution 
by means of the logical spectrum, that some of those terms 
could not be really indeterminate. Hventually I discovered 
a simpler method of finding the coefficients ; and the solution 
obtained by it may be verified by the spectrum. It consists 
in substituting the special values of a, b, c, d, e, f (due to sup- 
posing a particular term identical with the universe) in the 
original equations, and then solving for z and y. The values 
so found are the coefficients for the terms. 3 
For the first term abedef we get 
ety=l, z—-y=1; 
