210 
PROBLEM. 
[588 
respectively; and the like as regards the accented letters. Then A, B, C, F, G, H 
are the cosines of the angles which the edges of the tetrahedron ABGD subtend 
at 0; they are consequently the cosines of the six sides of the spherical quadrangle 
obtained by the projection of ABCD on a sphere centre 0; and they are therefore not 
independent, but are connected by a single equation; substituting for A, B, G, F, G, H 
their values, we have a relation between a, b, c, f, g, h, x, y, z, w; viz. this is the 
relation which connects the ten distances of the five points in space 0, A, B, G, D 
(and which relation was originally obtained by Carnot in this very manner). There is 
of course the like relation between the accented letters. 
The conditions as to the two tetrahedra are 
A=A', B = B', G=G\ F=F', G=G', H = H', 
which, attending to the relations just referred to and therefore regarding w as a given 
function of x, y, z, and w' as a given function of x, y', z\ are equivalent to five 
equations (or rather to a five-fold relation); the elimination of x, y', z' from the five 
fold relation gives therefore a two-fold relation between x, y, z, that is, between the 
distances OA, OB, OG; or the locus of 0 is as before a curve in space. 
The conditions may be written: 
y' 2 ¿2 _ 2Ay'z' = a 2 , x 2 + w' 2 — 2Fxw' = f' 2 , 
z 2 + x' 2 — 2Bzx' = b' 2 , y' 2 + w' 2 — 2 Gy'w' = g' 2 , 
x' 2 + y' 2 — 2Cx'y' = c' 2 , z' 2 + w 2 — 2Hz'w' = h' 2 ; 
whence eliminating x', y\ z, w, and in the result regarding A, B, G, F, G, H as given 
functions of x, y, z, w, we have between x, y, z, and w a three-fold relation determining 
w as a function of x, y, z, and establishing besides a two-fold relation between x, y, z. 
As a particular case: One of the tetrahedra may degenerate into a plane quadrangle, 
and we have then the problem: a given plane quadrangle ABGD being assumed to 
be the perspective representation of a given tetrahedron A'B'C'D', it is required to 
determine the positions in space of this tetrahedron and of the point of sight 0. 
A generalisation of the original problem is as follows: determine the two-fold 
relation which must subsist between the 4x6,= 24 coordinates of four lines, in order 
that it may be possible to place in the tetrad of lines a given tetrahedron; that is, 
to place in the four lines respectively the four summits of the given tetrahedron. It 
may be remarked that considering three of the four lines as given, say these lines are 
the loci of the summits A, B, C respectively, we can in 16 different ways place in these 
lines respectively the three summits, and for each of these there are two positions of 
the summit D; there are consequently 32 positions of D; and the two-fold relation, 
considered as a relation between the six coordinates of the remaining line, must in 
effect express that this line passes through some one of the 32 points.