richer in diphenylamine. This line was determined up to 120 atm. 
pressure. The significance of all the regions in which three-phase 
lines are absent can only be expressed by a series of p, z-diagrams. 
The above considerations foreshadow the possibility of enunciating 
in general terms the conditions for the existence of a solid phase in 
presence of one or two fluid ones, when traversing the region of 
the critical phenomena of those latter ones, also for those binary 
mixtures which in the liquid state are not miscible in all proportions. 
Mathematics. — “On non-linear systems of spherical spaces touching 
one another.” By Prof. P. H. Scnourr. 
1. Before passing to our real investigation it is necessary to find 
how many spherical spaces touch 7-1 spherical spaces given arbi- 
trarily in the 7-dimensional space S,. And in its turn the answer to 
this question demands a knowledge of the situation of the centres 
of similitude of those given spherical spaces. So we start with a 
study of these centres of similitude. To this end we represent the 
spherical space, which is in S, again the locus of the points situated 
at a distance 7 from the centre J/, by the symbol Sp, (M, 7). 
2. Just as is the case with two circles lying in the same plane, 
two spherical spaces Sp (J/,,7,) and Sp (M,,7,) lying in S, admit 
of two centres of similitude on the line M, J/, connecting the centres, 
an external one U, and an internal one /,,; through U, pass the 
lines P, P, connecting the extremities P,, P, of direct parallel rays, 
through /,, pass the lines P, P', connecting the extremities P,, pe 
of opposite parallel rays. 
Supposing that in S, a number of 7-+-1 spherical spaces Sp (Ml, 7), 
(k=1,2,...,n-+1) is given arbitrarily, we shall now investigate 
the situation of the (» +1), pairs of centres of similitude (U, 4) 
with respect to each other. To this end we first notice that the 
three pairs of centres of similitude of the three spherical spaces 
Spun (Mi, ri), @= 1, 2,3) form the three pairs of opposite vertices 
of a complete quadrilateral, each of the four triplets of points 
(U. U. U). (Ge Fn La) (1 U» 1). Am Ls U.) 
consisting of three points of a right line; we indicate these lines in 
the given order by 
12 
hee RAE en a 
. . . { 
If we now further regard the n—1 pairs of lines (/,2), Ie) through 
