578 ‘ 
[125.] Instead of inscribing a pyramid &’ in the pyramid 5, we may 
propose to exscribe to the latter a new pyramid a’B‘c’D’, or E‘, which 
shall be homologous with it, the given point & being still the centre of 
homology. In other words, the four new planes B cD’, . , ABYC’, or E,, 
E,, E,, Ep, are to pass: through the four given points A, B, Cc, D; and the 
four new lines aa‘, BB, cc, DD' are to concur, in the fifth given point B. | 
The solution of this problem i is found to be expressed by the reauess 
quinary symbols for the four sought planes : 
[e,] = [01113], . . [w,] = [11103]. 
In fact, the pyramid ’, with these four planes for faces is evidently ex- 
scribed to the pyramid axBcp, or E; and because its corners may be re- 
presented by these other quinary symbols, 
a’ = (30001), . . v' = (00081), 
the condition of concurrence is satisfied. We may remark that the plane 
E] of [123.] is the plane of homology of the two last pyramids © and 
E; and that this evseribed pyramid = is homologous also to the inscribed. 
pyramid zn’, the point = being still the centre, and the plane [=] ae 
plane of their homology. 
[126.] It may be remarked that the common trace of the two planes 
B, and D,, on the plane azc, is the line a’’B’c”; to construct, then, the 
exscribed pyramid &, we may construct the plane £, of one of its faces, by 
connecting the point p with the line 4”B/’c’; and similarly for the rest. 
Or if we wish to determine separately the new point, or corner, D‘, which 
corresponds to the given point D, we may do so, by the anharmonic eaten 
tion, 
(DD, ED‘) = 3; 
for which may be substituted* the system of the two following harmonic 
equations : 
(DD,EF) = (pp 'D,F) = - 1; 
where F is an auxiliary point, namely p,’. 
Parr VII.—On the Homography and Rationality of Nets in Space; and 
on @ Connexion of such Nets with Surfaces of the Second Order. 
[127.] In general, all geometric nets in space are homographie figures ; 
corresponding points, lines, and planes, being those which have the same 
(or congruent) quinary symbols, in whatever manner we may pass from 
one to another system of five initial points, a... 3; whereof it is still 
supposed that no four are complanar. All points, lines, and planes of 
any such Wet are evidently rational, in the sense [8.]| already defined, 
pa 
* Compare the note to [108.]. 
