164 
The first of these results shows that the effect of 7 upon any 
function of y alone is to change it into the like function of 
y+jx; and the effect of the same symbol upon any function of 
z alone is to change it into the like function of z+ kz. 
‘* But the second example shows that its effect upon yz is to 
change it, not into either (y +jx) (z+ ka), or (z+ha) (y+ja), 
but into half the sum of these different expressions. For 
4{(y+jau) (2 + kz) + (2+ ke) (y+ jx)| = yz + juz + hay, 
in virtue of the equation jk + kj = 0. 
‘‘ Again, the third example shows that the effect of 7 upon 
yz is to change it, not into any one of the three different ex- 
pressions, 
(y+ja)? (z+hz), (y+ ja) (z+khe) (yt+jx), and 
(z+ha) (y+jx), 
but into the third part of the sum of the three. It is easy to 
see that this result follows from the equations, 
pa-l, ‘B=-1, je+hj=0. 
‘«* Pursuing the same course we shall find that the effect of 
7 upon y*z* is to change it into the tenth part of the sum of 
the ten expressions, 
(y+je)? (2+ kx)’, 
(y+jx) (+h) (ytju) (z+ ka)’, 
(yt+jx) (z+ha)? (y+ju) (2+ he), 
(y+ ju) (2+hx) (y+ je), 
(z+hx) (y+ju) (z+ ke), 
(z+hax) (yt+jx) (z+ha) (y+ja) (2 +h), 
(2+hx) (yt+ja) (z+ka)? (yt+je), 
(z+hzx)? (y+ jx)? (2+ ke), 
(z+hu)? (yt jx) (2+ kv) (y+je), 
(e+ha) (y+ jz)’, 
which arise as the differently arranged products of the five 
factors, of which two are equal to y +jz, and three to 2+ Aa. 
‘‘ Following up the analogy, we are led to expect that the 
effect of + upon ¥” 2” will be to change it into the 
