» waite 
(599) 
RK EE TE CP 
the following theorems are proved in an entirely similar way: 
“The hyperspheres H,-; with n—z dimensions determined by 
“groups of n+ points of the curve N, conormal with n—2 given 
“points tj, tz, « « « « t—2 Of that curve, intersect the curve still in 
“n—1 fixed points sj, so, -.. sn-1 and form therefore a net, of 
“which the hypersphere H,,-3 determined by those »—1 points s is 
“the base. And if the system of the n—2 given points describes 
“the curve Vn, the groups of n—1 points s determine on N‚ an invo- 
“lution of degree n—1 with n—2 dimensions.” 
“If the point P describes the plane w common to the normal 
“spaces of n—2 given points t, fg, « « « « t—9, the centre M of the 
“corresponding hypersphere of JOACHIMSTHAL describes the plane ge, 
“which is the locus of the points at equal distance from the 
“,—1 points s depending in the indicated way on the n—2 given 
“points t; then P and M describe in the planes z and ye affinely 
“related point-fields (P) and (M).” 
«Between the spacial systems (P) and (M) with n dimensions 
“exists a correspondence (2n—1, 2n—1).” 
“The cyclographic representation of all the hyperspheres of 
«JOACHIMSTHAL demands the given space S, to be supposed to be 
“part of a Sn+1; it leads to a curved space of order 2 (2n—1) with 
“nx dimensions in this Sp+1 as locus of the pairs of images, etc.” 
We believe we can suffice with these general indications; we 
only wish still to observe that the coefficients of the equation deter- 
mining the 2—1 fixed points are connected in a simple way with the 
symmetrical functions = ¢ of the »—2 points t taken arbitrarily. 
n—2,k 
Mathematics. — “Approximation formulae concerning the prime 
numbers not exceeding a given limit”. By Prof. J. C. KLuvver. 
RieMann’s method for determining the total number of the prime 
numbers p less than a given nuinber c is equally serviceable, when 
it is required to evaluate other arithmetic expressions involving these 
prime numbers, for example the sum ~ ps of their (—s)" powers, 
pre 
or the least common multiple M(c) of all integers less than c. The 
different results, thus deduced, constantly contain a set of terms 
depending on the complex zeros “ of the Riemann ¢-function. The 
most important of these terms is always one and the same disconti- 
