On the Logical Spectrum. 287 
two possible classes are not represented ; hence the diagram 
is not general. 
Another method, which I propose to call the logical spectrum, 
is capable of representing quite generally the universe sub- 
divided by any number of marks. Let the universe be 
represented by a rectangular strip, as in diagrams 5 to 8. 
Diagram 5. Diagram 6. Diagram 7. 
a 
| 
~ Ss 
r SS AES te S = lhe 
|+| [=|]: S = : 
a'b’ 
abe' 
ab'e 
a'be 
abc! 
a'b'c 
a'b'c! 
Diagram 8. 
By subdividing into two we represent the possible classes 
formed by one mark a; by subdividing each of these parts 
again we represent the four possible classes formed by two 
marks a and ); and by subdividing each of these four parts 
again we represent the eight possible classes formed by the 
three marks a, b,c; and so on. This method aliows all the a 
part to be contiguous ; but the 6 part is broken up into two 
portions, the c part into four portions, the d part into eight 
portions. However, the regularity of the spectrum enables 
us easily to find all the portions belonging to any one mark. 
I shall apply this method to verify the solution of the logical 
equations 
Ufar+by=c}, 
Ufdz—ey=f} ‘ 
Here U is the symbol for the whole of the objects considered; 
it corresponds to the strip of paper in the diagram. The 
letters a, b, c, d, e, f represent known marks. There are two 
unknown marks denoted by x and y, which are such that the 
part of U having the marks a and «, together with the part 
having the marks b and y, is identical with the part having 
the mark c; and the part having the marks d and z, excepting 
the part having the marks e and y, is identical with the part 
having the mark f. It is required to select the part of U 
which has the mark wz, and ae the part which has the mark y. 
2 
