1892 - 93 .] On Division of Space into Lifinitesimcd Cubes. 195 
or 
\7 
y^qaq-'^^^Y^.qaq-'^ 
= V.( V qq-^qaq-'^ + qaq~~^ y qq~^) - 2S.ga2~^Vi.V5'i^7~l 
= 2qaq-~^^.^qq-^-2^.qaq~^^^.Yqpr^. 
From the sum of the three equations of this form (each multi- 
plied by its qaq~^) it ajDpears at once that 
s.y^g-^ = 0j 
so that, as qaq~^ may be amj unit vector, 
V.a^‘= -2SaVi.M-^ 
= <r'^ 
From two of the three equations of this form we have 
a,(v.a^.^) = 3.(v./J^.g); 
and this, by means of (5), gives 
-V.7(aa, + /33,)^ = ^y^‘, 
(0 
(5) 
or 
VI u ^ VJU d.M Vzi 
-Kr :r = ^ -:r 
u u u 
( 6 ) 
There are, of course, three equations of this form, and they give by 
inspection 
1 = „a V«« , V?« 1_ Vu 
V — = “3i^ 
u u u- 
yOg- 
-V 
3 id 
(7) 
The first and last of these equals give 
whose general solution is known to be 
Ld = %rr 
m 
^T(p-e)’ 
where m and « are constants. The other members of (7) show that 
