1892-93.] On Division of Space into Infinitesimal Cubes. 193 
Note on the Division of Space into Infinitesimal Cubes. 
By Professor Tait. 
(Read December 5, 1892.) 
Tlie proposition that “ the only series of surfaces which, together, 
divide space into cubes are planes and their electric images ” pre- 
sented itself to me twenty years ago, in the course of a quaternion 
investigation of a class of Orthogonal Isothermal Surfaces (Trans. 
R.S.E., Jan. 1872). I gave a second version of my investigation 
in vol. ix., p. 527, of our Proceedings. Prof. Cayley has since 
referred me to Note vi., appended by Liouville to his edition of 
Monge’s Application de V Analyse d la GeomUrie (1850), in which 
the proposition occurs, probably for the first time. The proof which 
is there given is very circuitous ; occupying some eight quarto pages 
of small type, although the reader is referred to a Memoir by Lame 
for the justification of some of the steps. But Liouville concludes 
by saying : — “ Tanalyse precedente qui etablit ce fait important n’est 
pas indigne, ce me semble, de I’attention des geometres.” He had 
previously stated that he had obtained the result “ en profitant 
d’une sorte de hasard.” As Liouville attached so much importance 
to the theorem, and specially to his proof of it, it may not be 
uninteresting if I give other modes of investigation. The first of 
them is merely an improved form of what I have already given in 
our Proceedings ; the second (which is the real object of this note) 
seems to have secured nearly all the advantages which Quaternions 
can afford, in respect alike of directness, clearness, and conciseness. 
It is very curious to notice that much of this gain in brevity is due 
simply to the fact that the Conjugate of a certain quaternion is 
employed along with the quaternion itself in my later work ; while 
I had formerly dealt with the reciprocal, and had, in consequence, 
to introduce from the first the tensor explicitly. The investigation 
should present no difficulties to anyone who has taken the sort of 
trouble to remember elementary quaternion formulse which every 
VOL. XIX. 22/12/92. X 
