(599 ) 
Deel ARD 
fr Aer anal AsO A a duns ie A, — uae =! = 
; WV, +r, +...+ 7, + nl)! 
Abt pl vl... pl 
aw; (» + n—l)! 
hence 
vl (nl)! 
po) M oe WEN OL ab ore (5) 
vp (» + n—1) 
( gn } Mael 
with 
pv, t+ry+...+m=vr. 
The sum to the right could evidently be arrived at out of 
(a, ta, +... a)’ by developing it according to the polynomial 
theorem and by suppressing all the polynomial coefficients. The factor 
v! (n—1)! 
(» + n—1)! 
is nothing else but the reciprocal of the number of terms. By introducing 
in 1) the obtained value of p”, we find, when the distance of the 
vertex a; of the simplex from the space £ or A is indicated by y;, 
of ' 
WG Eee M NS IND oP 
(»-+n—1)!l mm 
(vy, Hv, +..-4-m=?). 
For »=2 I have deduced (l.c) the sum in 3) to a sum of (n+41) 
respectively 7 squares, in other words I have substituted for the 
simplex a system of (x + 1) resp. m single material points, which 
is equivalent to it with respect to all questions connected with the 
moments of inertia. For » > 2 a similar reduction seems to be less 
easily effectible. 
Stuttgart, March 1905. 
Physiology. — “On the branchings of the nerve-cells in repose and 
after fatigue.’ By Dr. S. J. pe Langer. (Communicated by 
Prof. C. WINKLER.) 
In the laboratory of Marrutas Duvar some experiments have been 
made by MaroufrraN in order to ascertain whether it is possible 
to demonstrate modifications in the dendrites of the ganglion-cells in 
cases of sleep through fatigue. His results have been published in 
the “Comptes Rendus de la Société de Biologie, 28 Févr. 1898” and 
subsequently. 
The animals he made use of for his experiments were mice, and 
he proceeded in the following manner: For the space of an hour 
