1347 
the sum of the others, from which one finds that the angles have 
real values and that their sum is a. 
b. The plane PQR cuts both the-indicatrix and the conjugate 
indicatrix. In this case different positions of the triangle are still 
possible. We can however confine ourselves to triangles the three 
sides of which are real. These are really possible, for in the plane 
of a hyperbola we can draw triangles the sides of which are parallel 
to radius-veetors drawn from the centre to points of the curve (and 
not of the conjugate hyperbola). 
By a closer consideration of the triangles now in question it is 
found however that by the choice of our “natural” units one side 
is necessarily longer than the sum of the other two. Formula (4) 
then shows that the cosines of the angles are real quantities, greater 
than 1 in absolute value, two of them being positive, and the third 
negative. We must therefore ascribe to the angles imaginary or 
complex values. If for p >> +1 we put 
arc cos p = ilog (p + Vp? — 1) 
and 
are cos (— p) = 1 — are cos p , 
we find for the three angles expressions of the form 
. ta, ti? and w—-7(a + B), 
so that the sum is again a. — 
From the cosine calculated by (4) or (5) the sine can be derived 
by means of the formula 
sin p == V 1 — cos? , 
where for the case cos? p >1 we can confine ourselves to the value 
sin gp =iV cos? p —1 
with the positive sign. 
It deserves special notice that two conjugate radius-vectors of 
the indicatrix and the conjugate indicatrix are perpendicular to each 
other and that a deformation of the field-figure does not change the 
angle between two intersecting lines determined according to our 
definitions. 
$ 8. Before proceeding further we must now indicate the natural 
units ($ 5) for two-, three-, or four-dimensional extensions in the 
field-figure. Like the unit of length, these are defined for each 
point separately, so that the numerical value of a finite extension is 
found by dividing it into infinitely small parts. 
A two-dimensional extension cuts the conjugate indicatrix in an 
ellipse, or the indicatrix itself and the conjugate indicatrix in two 
86 
Proceedings Royal Acad. Amsterdam. Vol. XIX 
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