76 J 
SURFACES OF THE THIRD ORDER. 
451 
(x) 
a ± a 4 a 5 j 
(?) 
G-fi Cg> 
0) 
a 4 b 8 c 6 , 
(P) 
cq^Cg, 
(;v) 
MA, 
(g) 
b 2 c 4 a 5 , 
(in) 
biCzfis, 
(q) 
b 2 c 8 a 9 , 
(*) 
Ci C\ c 5 , 
(h) 
Cs®A, 
(n) 
c 4 a 8 b 6 , 
(?) 
c 2 af 9 , 
(0 
a 4 a 2 a 3 , 
(x) 
a a a 6 a 7 y 
a> 
(P/) 
a 2 b 6 c 7 , 
(v) 
MA, 
(y) 
b\ b 6 b 7 , 
( m /> 
b.fi 9 a 7 , 
A) 
b. 2 c 6 a 7 , 
(0 
Ci c 2 c 3 , 
( z ) 
c i c 6 c 7 , 
A) 
c 5 a 9 b 7 , 
( r /) 
c,a,,b 7 , 
(x) 
cq 0-8^9, 
(l) 
A) 
C^sb 9 C 8 , 
(y) 
bi b s b 9 , 
(A) 
b 4 c s a 8 , 
A) 
b 3 c a a 8) 
(z) 
Ci c 8 c 9 , 
A) 
c 4 gA j 
A) 
c 3 a a b 8 . 
The preceding method was the one that first occurred to me, and which appears 
to conduct most simply to the actual analytical expressions for the forty-five planes; 
but it is worth noticing that the relations between the lines and planes might have 
been obtained almost without algebraical developments, if we had supposed that P, 
instead of representing a proper surface of the second order, had represented a pair 
of planes. This would have conducted at once to one of the one hundred and twenty 
forms U, e.g. U = w66 + k%7/%. Or changing the notation so as to include k in one of 
the linear functions, U = ace — bdf and it is indeed obvious d priori, by merely reckoning 
the number of arbitrary constants, that any function of the third order can be put 
under this form. If we suppose cl = pb to be the equation of one of the triple tangent 
planes through the intersection of the planes a and b, the plane a = pb meets the 
surface in the same lines in which it meets the hyperboloid ¡ice — df= 0, that is, the two 
lines in the plane are generating lines of different species, and consequently one of 
them meets the pair of lines cd and ef, and the other of them meets the pair of 
lines cf and de (where cd represents the line of intersection of the planes c = 0, d — 0, 
&c.). This suggests a notation for the lines in question, viz. each line may be repre 
sented by the three lines which it meets, or by the symbols ab.cd.ef and ab.cf.de. 
Or observing that ¡jb has three values, and that the same considerations apply mutatis 
mutandis to the planes through be and ca, the whole system of lines may be repre 
sented by the notation, 
ob, 
ad, 
af 
cb, 
cd, 
cf, 
eb, 
ed, 
cf, 
(ab. cd. ef) u 
(ab. cd. ef\, 
(ab. cd. ef) 3 
(ad. cf. eb ) u 
(ad. cf. eb ) 2 , 
(ad. cf. eb ) 3: 
(af. cb .ed ) x , 
(af. cb .ed\, 
(af. cb .ed) 3 
(ab.cf. ed 
(ab. cf. ed ) 2 , 
(ab. cf. ed ) s 
(ad. cb. ef\, 
(ad. cb. ef) 2 , 
(ad. cb. ef\ 
(af. cd .eb\, 
(af. cd .eb) 2 , 
(af. cd. eb ) 3 
57—2