154 
Now let x be a category of simplicial divisions of G analogous 
to yw. The resulting simplicial divisions of g belong to a category w 
of the kind just described. Let G',G",... be a sequence of indefini- 
tely condensing simplicial approximations of (# corresponding to z, then, 
at the same time, there is hereby determined a sequence g', g’,... 
of indefinitely condensing simplicial approximations of g corresponding 
to w. Since, in virtue of III, the value of (1) over g® is equal to 
the value of (2) over G, there exists, just as there does a value 
of (1) over g for w, a value of (2) over G for x, both values being 
equal, and not changing if some other category of the same kind is 
chosen in the place of either or x. 
wy 
In introducing le. p. 70 the notion of a second derivative, we have 
omitted to give the definition of the underlying concept of normality 
of an S, provided with an indicatrix and an Seay provided with 
an indicatrix which are perpendicular to each other in an 5S, pro- 
vided with an indicatrix. This definition we shall here give. 
Let 7 be the point of intersection of Sp and S,_,, «,'...a, T 
the indicatrix of S, and 8,...8,-,7' the indicatrix of S,_,; we call 
Sp normal to Sep and’ Sj postnormal 10. )5,,.0 ere ip, ann 
is an indicatrix of S,. 
Thus, for some values of » the concepts normal and postnormal 
are equivalent, for other values not equivalent. 
Furthermore we call a p-dimensional vector system V normal to 
an (n—p)-dimensional vector system W at the same point, and 
W postnormal to JV, if, with respect to a rectangular system of 
coordinates the components of V are respectively normal to and of 
equal scalar values as the components of JW. 
In this terminology, the second derivative of the vector distribution 
a —p pv’ 
X is the normal distribution of tbe first derivative of the postnor- 
mal distribution of ’_X. 
