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} ·---------------· ·-------· | | | | | | | ·---|---· | |---|---| | |(O)|(O)| | |(I)| | |---|---|---|---| ·-------· | |(I)| | | | ·---|---· | .'. x'y_{1} | | | ·---------------· In this case we see that the Conclusion is an Entity, and that the Nullity-Retinend has changed its Sign. 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 x'm_{0} + ym_{1} (which ¶ xy_{1}) x_{1}m'_{0} + y'm'_{1} (which ¶ x'y'_{1}) m_{1}x_{0} + y'm_{1} (which ¶ x'y'_{1}).] The Formula, to be remembered, is, xm_{0} + ym_{1} ¶ x'y_{1} with the following Rule:-- _A Nullity and an Entity, with Like Eliminands, yield an Entity, in which the Nullity-Retinend changes its Sign._ [Note that this Rule is merely the Formula expressed in words.] pg077 Fig. III. This includes any Pair of Premisses which are both of them Nullities, and which contain Like Eliminands asserted to exist. The simplest case is xm_{0} + ym_{0} + m_{1} [Note that "m_{1}" is here stated _separately_, because it does not matter in which of the two Premisses it occurs: so that this includes the _three_ forms "m_{1}x_{0} + ym_{0}", "xm_{0} + m_{1}y_{0}", and "m_{1}x_{0} + m_{1}y_{0}".] ·---------------· ·-------· | | | | | | | ·---|---· | |---|---| | |(O)|(O)| | | |(I)| |---|---|---|---| ·-------· | |(O)|(I)| | | ·---|---· | .'. x'y'_{1} | | | ·---------------· In this case we see that the Conclusion is an Entity, and that _both_ Retinends have changed 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