theorem to determine whether a solution of a differential 
equation is singular or particular, and an attempt is made 
to solve this problem without using a troublesome transfor- 
mation as has been done by Boole and others. 
Ordinary Meeting, November 28th, 1882. 
J. P. Joule, D.C.L., LL.D., F.B.S., &c., Vice-President, in the 
Chair. 
“ On the Transformation of a Logical Proposition contain- 
ing a Single Kelative Term,” by Joseph John Murphy, 
F.G.S. Communicated by the Bev. Robert Harley, F.R.S. 
Abstract. 
In the system here proposed, R means any relation what- 
ever, and R~ l the inverse relation. The equation 
X=RY 
and its inverse 
Y = R~ 1 X 
mean respectively that X is teacher of Y and Y is pupil of 
X, without implying whether or not X has any other pupils 
and Y any other teachers. 
X = 1RY 
and its inverse 
Y=1~ 1 R~ 1 X 
mean respectively that X is the only teacher of Y, and Y 
the pupil of none but X. 
If X and Y are the names of classes instead of individuals, 
1X< JRY 
means that every X is teacher of a Y : and its converse 
ij? -1 lX< Y 
means that pupils of every X are included in the class Y 
The proposition 
1X< RIY 
