G. PLARR ON THE SOLUTION OF THE EQUATION Vpgp= Oy, 81 
It will now be evident that the condition '> 0 will be- satisfied by every 
value of « whose extremity is comprised within the inside of the surface '=0 
taken in its whole ; and in particular, that condition will be satisfied, for the 
proper values of ¢, when the direction of « is such as to intersect the first, 
second, and third sheets at the same time; and this is due to the circum- 
stance that I changes sign whenever, Ue remaining the same, the varying 
tensor ¢ is such as to engender an intersection of e by the surface. 
If we now try to describe the general aspect of the surface T = 0, 
bp = Reg 
2=hs 4 
when h, > 0, in looking at the surface from a point in the plane Ap, it would 
present a kind of continuous belt, formed by the third sheet, and whose axis . 
(in a certain vague sense of axis) is the axis v. 
The belt would have its greatest breadth in the plane Xv, and its smallest in 
the plane pv, because the upper edge of it is formed by the spherical conic 
whose equations are 
we may say (excepting the extreme cases of or h, = 0) that 
the cone 0 = LhS*re’ . 
and the sphere Teo=¢=*./=P ; 
so: that the belt, as seen from the origin 0, would subtend the double angle 
whose tangent: is 
in the plane hv, tke a =), 
a Te BNE © Nh 
: Ze Sve aN 
in the plane py, tgpe,, = Sue = — , 
3 
both real because h; < 0, and the first greater than the second because of 
fi > “he . 
Tf we would trace or inscribe on the outer surface of the belt the spherical 
conics, we would find them belonging to two different sets, divided mto two 
corresponding regions by the two circles of radius ¢, and whose planes, R, R’, 
passing through the axis jw, are symmetrical on both sides of the plane pv. 
In the part of the surface comprised in the angle of the two planes R, R’, 
which contains the plane \y» and the axis i, the conics would in a certain 
sense be concentric about the central point ¢., on the axis X. 
In the part of the surface of the belt comprised between the above two 
planes and in the angle which contains the plane py, the conics belong to the 
set which has their quasi centre in the axis v, in ¢,v. 
VOL. XXVIII. PART I. 7 Pie 


