454 
ON THE TRIPLE TANGENT PLANES OF 
[76 
which remains unaltered for cyclical permutations of l, m, n, i.e. the anharmonic ratio 
of x, x, x is the same as that of y, y, y, y, or z, £ z, z; there is of course no 
correspondence of x to y or £ to y, &c., the correspondence is by the general pro 
perties of anharmonic ratios, a correspondence of the system x, f, x, x, to any one of 
the systems (y, y, y, y), or (y, y, y, y), or (y, y, y, y), or (y, y, y, y), indifferently. 
The theorems may be stated generally as follows: “ Considering two lines in the same 
triple tangent plane, the remaining triple tangent planes through these two lines 
respectively are homologous systems.” 
Suppose the surface of the third order intersected by an arbitrary plane. The 
curve of intersection is of course one of the third order, and the positions upon this curve 
of six of the points in which it is intersected may be arbitrarily assumed. Let these 
points be the points in which the plane is intersected by the lines a x , b x , a 6 , b 6 , c 6 , a 8 ; 
or as we may term them, the points a x , b x , a 6 , b 6 , c 6 , a s .Q) The point c x is of course the 
point in which the line a 1 b 1 intersects the curve. The straight lines a 4 b 6 c 8 , b 4 c 6 a 8 , c 4 a 6 b 8> 
and a 4 b 8 c 6 , b 4 c 8 a 6) c 4 a 8 b 6 , show that c 4 and h A are the points in which a 8 b 6 , and a 8 c 6 inter 
sect the curve, and then b 8 and c 8 are determined as the intersections of a 6 c 4 , a 8 b 4 with 
the curve. The intersection of the lines b 6 c 8 and b 8 c 6 (which is known to be a point 
upon the curve by the theorem, every curve of the third order passing through eight 
of the points of intersection of two curves of the third order passes through the 
ninth point of intersection) is the point a 4 . The systems a 4> b 4 , c 4 ; a 6 , b 6 , c 6 ; a 8 , b 8 , c 8 , 
determine the conjugate system a 5) b 5> c 5 ; a 7 , b 7 , c 7 ; a 9 , b 9 , c 9 ; by reason of the straight 
lines a x a 4 a 5 , b x b 4 b 5 , c x c 4 c 5 ; a x a 6 a 7 , b x b 6 b 7 , c x c 8 c 7 \ a x a 8 a 9 , bj) 8 b 9 , c x c 8 c 9 , viz. a 5 is the point where 
a 4 a 4 intersects the curve, &c. The relations of the systems (a 4 , b 4 , c 4 ; a 5 , b 5 , c 5 ), (a 6 , b 6 , c«; 
a 7 , b 7 , c 7 ), (a 8 , b 8 , c 8 ; a 9 , b 9 , c 9 ) to the system a 4 , b x , c 4 ; a 2 , b 2 , c 2 \ a 3 , b 3 , c 3 are precisely 
identical. It is only necessary to show how the points u 2 , b. 2 , c 2 ; a s , b 3 , c 3 of the 
latter system are determined by means of one of the former systems, suppose the 
system a 6 , b 6 . c 6 ; a 7 , b 7 , c 7 ; and to discover a compendious statement of the relation 
between the two systems. The points a 4 , b 4 , c 4 ; a 2 , b 2 , c 2 ; a 3 , b 3 , c 3 ; a 6 , b 6 , c 6 ; a 7 , b 7) c 7 , 
are a system of fifteen points lying on the fifteen straight lines a 2 b 2 c 2 , a 3 b 3 c 3 , 
a 4 a 2 a 3 , b 4 bjj 3 , c x c 2 c 3 , a 4 a 6 a 7 , b } bjj 7 , c 4 c 8 c 7 , a 3 b 7 c s , b 3 c 7 a 6 , c 3 ci 7 b 6 , a 2 b 6 c 7 , b 2 c 6 a 7) c 2 a 6 b 7 , viz. the nine 
points a x , b 4 , Ci, a 2 , b 2 , c 2 ; a 3 , b 3 , c 3 are the points of intersection of the three lines 
a 4 biCi, a 2 b 2 c 2 , a 3 b 3 c 3 with the three lines a x a.fl 3 , b x h.b 3 , c x c 2 c 3 , and the remaining six points 
form a hexagon a 6 b 7 c 6 a 7 b 6 c 7 , of which the diagonals a B a 7 , b 6 b 7 , c 8 c 7 pass through the points 
CLi, b x , Ci, respectively, the alternate sides a 6 b T , c 6 a 7 , and b 6 c 7 pass through the points 
c 2 , b 2 , a 2 respectively, and the remaining alternate sides b 7 c 6) a 7 b 6 , and c 7 a 6 pass through 
the three points a 3 , b 3 , c 3 respectively. The fifteen points of such a system do not 
necessarily lie upon a curve of the third order, as will presently be seen: in the 
actual case however where all the points lie upon a given curve of the third order, 
and the points a x , b x , c x ; a 6 , b 6 , c 6 ; a 7 , b 7 , c 7 are known, a 2 , b 2 , c 2 ; a 3 , b 3 , c 3 are the 
intersections of the curve with b 6 c 7 , c 8 a 7 , a 6 b 7 , b 7 c 6 , c 7 a 6 , a 7 b 6 respectively, and the fact 
1 In general, the point in which any line upon the surface intersects the plane in question may be repre 
sented by the symbol of the line, and the line in which any triple tangent plane intersects the plane in question 
may be represented by the symbol of the triple tangent plane: thus, a x , b x , c x are points in the line a 1 6 1 c 1 , or in 
the line w, &c.