1892 - 93 .] On Division of Space into Injinitesimcd Cubes. 197 
so that 
Ug 
• U5' = y 03 -^ = /3a. 
'Tq 
1 U<? 
3'^ Tq 
(0 
Thus, as the form of the three middle terms shows that their 
common value must be some constant quaternion, 
or 
Ug Ug 
Ug 
r^^-pa 
for we need not add a constant vector to p, and the form of the 
first of the five equal quantities above shows that no quaternion 
constant (except, of course, one of the form ea already referred to) 
can be added to the right-hand side. 
Thus, finally, as before 
d(T ~ - ap ^dpp , 
Though the methods employed in these two investigations are, at 
least at first sight, entirely different, it will be easily seen that the 
equations (7) and (6) to which they respectively lead are identical in 
meaning with one another, term by term. Yet the former shows 
two differentiations in every term, while the second appears to in- 
volve one only. Thus also, two distinct integrations were required 
in the first solution, while one sufficed for the second. But in 
the first, the tensor and versor of the quaternion were all along 
separated ; in the second the quaternion itself was directly sought. 
