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PROBLEMS AND SOLUTIONS. 
viz., the line AJ passes through a, the line AK through d, &c.; the proof that AJ 
passes through a depends on the identical equation 
x, y, z 
x, z, y 
0, 1,-1 
and the like for the other lines AK, AL, &c. 
The lines through the tangentials are the 36 lines obtained by joining any two 
of the points (A, B, G, D, E, F, G, H, I) and the 36 lines obtained by joining any 
two of the points (J, K, L, M, N, 0, P, Q, R) ; and these 72 lines pass through the 
tangentials, as shown by the table 
ABC, 
BDI, 
CEG, 
JKL, 
KMR, 
LXP, 
ADG, 
BEH, 
GFI, 
JMP, 
KNQ, 
LOR, 
AEI, 
BF G, 
DEF, 
JNR, 
JOP, 
MNO, 
AFH, 
GDH, 
GHI, 
JOQ, 
LMQ, 
PQR, 
viz., in the triad ABC, BG passes through A', GA through B', AB through G'; and 
the like for the other triads. The proof that BG passes through A depends on the 
identical equation 
y z x =0; 
z , x , y 
x(x? — z 3 ), y (z 3 — ¿e 3 ), z (oc 3 — y 3 ) 
and the like for the other combinations of points. 
If we attend only to the points A, B, G and their tangentials A', B', C'; then 
we have on the cubic three points A, B, G, such that the line joining an} r two of 
them passes through the tangential of the third point. And the figure may be con 
structed by means of the three real points of inflexion a, d, g, as follows, viz., joining 
these with any point J on the cubic, the lines so obtained respectively meet the cubic 
in three new points which may be taken for the points A, B, G. Or if one of these 
points, say A, be given, then joining it with one of the three real inflexions, this 
line again meets the cubic in the point J, and from it by means of the other two 
real inflexions we obtain the remaining points B and G; it is clear that, A being 
given, the construction gives three points, say J, K, L, each of them leading to the 
same two points B and G. 
We may consider the question from a different point of view. Let A, B, G be 
given points, and let there be given also three lines passing through these three 
points respectively; through the given points, touching at these points the given lines 
respectively, describe a cubic; and let the given lines again meet the cubic in the 
points A', B', C' respectively. The equation of the cubic contains three arbitrary 
C. V. 76