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TRANSACTIONS OF SECTION A. 401 
The equation of any n-hypersphere is homogeneous of the first degree, viz. :— 
U FMB, + Oey ves | Ap yo Enzo = O,; 
where a, is the cosine of the angle at which this hypersphere intersects the Kth 
fundamental hypersphere. 
The radius p is given by the formula 
Lee 
i. Pe 
The co-ordinates of the centre are given by 
= (P24 Pe), 
Pr 
ww 
Thus 
n 
2, ‘pe, = (n—2r—1) PrizPny2 
represents an »-sphere which intersects every r-face of the figure, whose vertices are 
the first n--1 of the fundamental points, in a section whose diameter is the line joining 
the orthocentre and centroid of that face. The radius p, is given by 
4(r +1) p2= rie p72 -(n — 2r — 1)? p. 
. 
ut2 
ai 
The centre, O,, for all values of 1, lies on the line joining O, the circumcentre, to P, 
the orthocentre, so that 
PO, = PO/r+1. 
If 2r=n, the equation becomes symmetrical with respect to all the fundamental 
spheres, and gives the complete analogue to the nine-point circle of plane geometry. 
The homogeneous equation of the second degree is in general that of a locus 
analogous to bicircular quartics, or to cyclides, reducible to analogues of conic sections 
or conicoids. 
TUESDAY, SEPTEMBER 16. 
Joint Meeling with Section .—See p. 553. 
DEPARTMENT OF GENERAL PuysiIcs. 
The following Papers were read :—- 
1. The Electric Arc as a Standard of Iaght. By J. F. Forrest. 
» 
2. Some Experiments in Contacts between Klectrical Conductors. 
By Dr. W. H. Eccuss. 
When a current is passed across a ‘loose contact’ the relation between the 
applied electromotive force and the current produced is, in general, not a linear 
one, and no sufficient explanation of the observed phenomena has hitherto been 
offered. The present author investigates whether the behaviour of contacts can 
be accounted for by purely thermal actions in the matter near the contact. 
The Joule, the Peltier, and the Thomson effects will all play a part, and the 
alterations of resistivity with temperature, as well as alterations of the geo- 
metrical configuration of the surfaces in contact caused by thermal expansion, 
ought all to be taken into account. These thermal effects are, of course, most 
noticeable in contacts between bad conductors of electricity and of heat, as, 
e.g., the natural crystalline oxides and sulphides. 
It is advantageous to separate contacts into two classes: First, those between 
like substances; second, those between unlike substances. In the first class there 
is in general no thermoelectric action, and the non-linearity of the relation 
1913, DD 
