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+ ka. 
‘‘ But the second example shows that its effect upon yz is to 
change it, not into either (y + jx) (z+ ka), or (z+hx) (y+ja), 
but into half the sum of these different expressions. For 
3{(y+ja) (e+ kz) + (2+ ke) (y+jx)} = yz + juz + hay, 
in virtue of the equation jh + 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+ je) (z+he) (y+jz), and 
(z+khx) (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, 
poH-l, ‘R=-1, jk+kj=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+ju) (2+ ka)’, 
(y+jx) (e+hkx) (ytjau) (2 + kx)’, 
(y+ja) (2+ha)? (y+ju) (2+ ha); 
(y+ju) (2+ hu) (y+ je), 
(z+hx) (y+ju) (2+ ke)’, 
(z+hx) (y+ju) (z+hu) (ytjax) (2+ha), 
(2 +ha) (y +jo) (2+ ke)? (y + je), 
(z+khz)? (y+ ju)? (2+ ke), 
(z+khx)? (yt+jx) (2+ kha) (y+ja), 
(etka) (y+ ja)’, 
which arise as the differently arranged products of the five 
factors, of which two are equal to y +jx, and three to z + ha. 
“‘ Following up the analogy, we are led to expect that the 
effect of 7 upon y” 2” will be to change it into the 
