useful Formulae. I shall call them "Fig. I", "Fig. II", and "Fig. III".
pg075
Fig. I.

This includes any Pair of Premisses which are both of them Nullities,
and which contain Unlike Eliminands.

The simplest case is

.---------------. .-------.
xm_{0} + ym'_{0} |(O) | | |(O)| |
| .---|---. | |---|---|
| |(O)|(O)| | | | |
|---|---|---|---| .-------.
| | | | |
| .---|---. | .'. xy_{0}
|(O) | |
.---------------.

In this case we see that the Conclusion is a Nullity, and that the
Retinends have kept their Signs.

And we should find this Rule to hold good with _any_ Pair of Premisses
which fulfil the given conditions.

[The Reader had better satisfy himself of this, by working out,
on Diagrams, several varieties, such as

m_{1}x_{0} + ym'_{0} (which > xy_{0})
xm'_{0} + m_{1}y_{0} (which > xy_{0})
x'm_{0} + ym'_{0} (which > x'y_{0})
m'_{1}x'_{0} + m_{1}y'_{0} (which > x'y'_{0}).]

If either Retinend is asserted in the _Premisses_ to exist, of course it
may be so asserted in the _Conclusion_.

Hence we get two _Variants_ of Fig. I, viz.

(a) where _one_ Retinend is so asserted;

(b) where _both_ are so asserted.

[The Reader had better work out, on Diagrams, examples of these
two Variants, such as

m_{1}x_{0} + y_{1}m'_{0} (which proves y_{1}x_{0})
x_{1}m'_{0} + m_{1}y_{0} (which proves x_{1}y_{0})
x'_{1}m_{0} + y_{1}m'_{0} (which proves x'_{1}y_{0} + y_{1}x'_{0}).]

The Formula, to be remembered, is

xm_{0} + ym'_{0} > xy_{0}

with the following two Rules:--

(1) _Two Nullities, with Unlike Eliminands, yield a Nullity,
in which both Retinends keep their Signs._
pg076
(2) _A Retinend, asserted in the Premisses to exist, may be
so asserted in the Conclusion._

[Note that Rule (1) is merely the Formula expressed in words.]

Fig. II.

This includes any Pair of Premisses, of which one is a Nullity and the
other an Entity, and which contain Like Eliminands.

The simplest case is

xm_{0} + ym_{1}

.---------------. .-------.