(596 ) 
another way in the “Rendiconti del Circolo Matematico di Palermo”, 
Vol. XIX (1905) and has arrived at the same result. Instead of 
contenting myself with the reference to these facts 1 wish to com- 
municate how in the same way moments of any higher order than 
the second can be found. It is true this problem has been prepared 
in the above mentioned place, pages 146—147, for a simplex so far, 
that but a slight step would have been necessary to bring about its 
solution. 
Let us presuppose a flat space of (n — 1) dimensions, a space of 
“rank” (“Stufe”) 7 as GRASSMANN expressed it as early as 1844, or of 
“point-value’ 2 as Prof. Scrovrr has said in his excellent textbook 
on polydimensional geometry. The moment J/, of order v of an 
arbitrary material figure belonging to this space with respect to a 
space ME of rank (n—1) (thus n—2 dimensions) contained in the 
same space is 
i f r’ dm, 
where 7 indicates the distance of a central point p in an element 
of that figure from ZE, dm the mass of the element. According to 
GRASSMANN however 
7 =[Ep], 
i.e. equal to the “outer” product of ZE and p, when we assign 
both to # and p the numerical value 1, consequently 
M, = { Eph a . Mes oes 
I assume that rv is a positive integer. If » is an even number 
and if the moment is to be calculated with respect to a space A of 
a rank smaller than (n — 1), if thus it is e.g. a case of a moment 
of inertia with respect to an axis (yp=2), then according to GRASSMANN 
x? = [Ap| Ap], 
where the symbol | denotes the “inner” multiplication, and we 
arrive at 
v 
2 
M,= | [Ap|Ap]adm. . TED) 
The integrals appearing in a) and 6) can be evaluated by one 
and the same integration, if we make use of the very useful notion 
of the “gap-expressions’ introduced by GRASSMANN. If namely we 
place the point p appearing in [Zp}’ or in [Ap | Ap]? symbolically 
