of Edinburgh, Session 187 8-7 9. 
61 
high, as that of the water a few inches above the bottom was 38 -5° ; 
the same water at No. 3 being 37 ’4° and the mud 37*8°. No. 1, 
being on the slope next the town, is exposed to the filth to be 
derived from it ; it is therefore not surprising that the bottom water 
is warmer than at No. 3, which is situated on the banks near the 
northern shore, where the water is comparatively clear. 
On the Principles of the Logical Algebra ; with Applications. 
Part II. By Alexander Macfarlane, M.A., D.Sc. 
(Read 20th January 1879.) 
{Abstract.) 
The equation 
x 2 = x 
expresses the condition that the symbol x be single and positive. 
The equation 
x 2 — —x 
expresses the condition that the symbol x be single and negative. 
The principle of contradiction 
x (1 - x) =0 
can be legitimately deduced with the help of the fundamental 
axioms of the science. 
The memoir contains a general investigation of the conclusions 
which can be drawn from two or more premises : — 
(1) of the form 
x = m; 
(2) of the form 
xy = m; 
(3) of the form 
x = y + z ; 
and investigates the fundamental relations which subsist between 
single functions of any number of independent symbols. 
VOL. X. i 
