106 Proceedings of the Royal Society 
circular line ; those having both x and y by the part inside both 
lines, those having neither by the part outside. 
vr xy 
JNow xy = xy . \ y = — - . 
Expand — in terms of the parts formed by means of xy and x. 
x 
XV 
= a xy x + h xy (1 - x) + c(l - xy)x + d(l - xy)( 1 - x ) . 
Let xy — 1 and x — \. Then, a = = 1 . 
xy = 1 and x — 0, then b = ^ • 
xy = 0 and x — 1, then c = ^ = ^ • 
xy = 0 and a? = 0, then d = ^ . 
. ’ °~x = XyX + 0(1 “ a#)* + 5(1 -&y)(l - #), 
and also xy(l — x) = 0 , as evidently ought to he the case. 
By putting in 
x 2 = x , or x(\ - x) = 0 , 
wo get ^ = xy + 0 ^(1 -y) + (1 - x ) , 
that is, what is y is identical with what is x and y, together with 
no part of what is x and not y , together with an indefinite part 
of what is not x. The truth of this is evidenced by the diagram. 
Since every logical equation is true arithmetically, 
y = «y + [j (! - ^ ; 
where the bar denotes that the arithmetical value of the symbol is 
taken. 
Applications of the above Theorem. 
(1.) The probability that it thunders upon a given day is£>, the 
probability that it both thunders and hails is q, hut of the connec- 
tion of the two phenomena of thunder and hail, nothing further is 
supposed to he known. Required the probability that it hails on 
the proposed day. 
