GENERAL EQUATION OE A CUBIC SURFACE. 
G5 
common, there are twenty-two triple tangent planes which have no line in common 
with the first two planes. 
This result may be obtained by counting the triple tangent planes which do not 
contain any of the six lines 4, 5, 6, 7 , 8 or 9, or it may be calculated otherwise. 
§ 20. But among these twent^^-two planes, there are three distinct tyj)es of 
relationship to the first pair of planes. 
The only type with which we are here concerned, is that in which the first line of 
the third plane cuts the first line of the second and of the third planes; the second 
line cuts the second lines, and consequently the third line cuts the third lines 
In this case, the first lines form a triple tangent plane, as do also the second lines 
and the third lines. 
In Figure B it is easily seen that the nine lines 
4 , 6,5 
9,8,7 
13 , 10 , 3 
give triple tangent planes when the numbers are read either horizontally or verticallju 
The only B triangles which do not contain any of the lines 4, 6, 5, 9, 8, 7, 13, 10, 
3, are as follows :— 
we must take also 
and 
1 
5 
16 
} 
19 
1 
J 
17 
3 
18 
2 
) 
21 
5 
22 
Q 
) 
20 
3 
23 
11 
5 
22 
3 
27 
11 
3 
23 
3 
26 
12 
J 
18 
3 
25 
12 
3 
19 
3 
24 
14 
5 
16 
8 
25 
14 
3 
17 
3 
24 
15 
3 
21 
3 
26 
15 
3 
20 
> 
27. 
of triangh 
3S we 
select ' 
1 
3 
16 
5 
19, 
12 
> 
18 
3 
25 
14 
3 
17 
5 
24, 
K 
MDCCCXCIV.—A. 
