(597 ) 
outside the brackets and if with PEANo we indicate every gap thus 
formed by +, we arrive at 
v 
r— |Help» resp. % == [As|As|?’. p 
or 
pe ahh 
where the expression Zi (furnished with » gaps) is equal to |Z} 
v 
in the first case and to [A+lA+|? in the second. The expression Z 
remaining constant in the integration it can be placed before the 
symbol f of the integral, so that we get 
M, = b.{p dm = Ip Hulsen hee arete “ate UL) 
This has reduced our problem to the determination of the “point- 
quantity of order v” 
pe) = frr an (2) 
belonging to the given material figure. (The vt power of a point 
p we have to imagine as the v-fold point p. The algebraic product 
of » different points is the total of these points, where on account 
of the interchangeability of the factors of an algebraic product the 
order of succession of the points is arbitrary. The sum of an arbitrary 
number of such like quantities has primarily but a formal meaning, 
but then it may be represented geometrically by a figure of order 
vy, the analogon of the ellipsoid of inertia). The integral 2) is depen- 
dent only on the form and the distribution of the mass of the given 
material figure, and whilst when treating our problem in the usual 
way with the aid of cartesian coordinates the space or A may 
have a very disturbing influence upon the integration this influence 
is here entirely done away with. Various other problems lead to a 
similar integral as 2). If inter alia we wish to calculate the kinetic 
energy 7’ of an (invariable or affinitely variable) continuously moving 
system of masses for an arbitrary epoch, then 
2 T = [v%dm, 
where v denotes the velocity of a central point p in the element 
: : 3 : Í; 
dm. But v? is equal to the “inner” square of the vector — repre- 
at 
senting the velocity of p according to length and direction, i.e. 
41* 
