ON THE SKEW SURFACE OF THE THIRD ORDER. 
93 
[322 
r -nZ)) 
generating line 
n the equation 
- nZ) 3 } = 0 ; 
ie and the line 
7 — nZ) 3 } 
<S>(X, F, Z), 
tting the factor 
322] 
which is the equation of the surface. It only remains to find <i>: writing for this 
purpose X + mZ, Y + nZ in the place of X, Y, respectively, and putting for a moment 
<t>(X + mZ, Y+nZ, Z) = <$>', 
we have 
(a 3 + /3 3 ) XYZ 0mY- nZ) - a/3 {/3 (X + mZ) - a (F + nZ)} (X 3 + F 3 ) = (/3X — aF) <f>'; 
that is 
(/3X - aF) d>'= £ {(a 3 + /3 3 ) XY(mY - nZ) - (X 3 + F 3 ) a/3 (m/3 - na)} - (/3X - aF) a/3 (X 3 + F 3 ); 
or, effecting the division, 
d>' = Z {(X 2 a — F 2 /3) (an — /3m) — X F (a 2 m 4- fi 2 n)} — a/3 (X 3 4- F 3 ), 
and then writing X — mZ, F — nZ in the place of X, F respectively, we have 
(X, Y, Z) = Z {((X - mZf a-(Y- nZf ¡3) (an - /3m) 
- (X - mZ) (Y-nZ) (a?m + /3 2 n)} - a/3 {(X - mZf + (F - nZf). 
Hence, finally, the equation of the surface is 
(a 3 + /3 3 ) (X — mZ) (Y — nZ) — a/3 {(X - mZ) 3 + (Y — nZ) 3 } 
+ £ {((X - mZ) 3 a -(Y-nZ) 3 /3) (an - /3m) - (X - mZ) (Y - nZ) (a-m + &n)} = 0, 
which is, as it should be, of the third order. 
Arranging in powers of Z and reducing, the equation is found to be 
(a 3 + /3 3 ) XY — a/3 (X 3 + F 3 ) 
+ Z {- (a 3 + /3 3 ) (?nY+ nX)+ (X-a + Y-/3) (m/3 + na) + a/3 (mX 2 + nY 2 ) - (a 2 m + f3 2 n) XY} 
+ Z 2 [mn (a 3 + /3 3 — a 2 X — /3 2 F) (¡3n z — am 2 ) (¡3X — aF)} = 0. 
The first form puts in evidence the nodal line 
(X — mZ = 0, Y —nZ = 0), 
and the second form puts in evidence the simple line 
( X - a = 0, F -/3 = 0). 
But to obtain a more convenient form, write for a moment X — mZ=P, Y — nZ=Q; 
the equation is 
(a 3 + /3 3 ) PQ - a/3 (P 3 + Q 3 ) + Z {(P 2 a - Q 2 /3) (na - m/3) - PQ (ma 2 + n/3 2 )} = 0, 
01-, as this may be written, 
(a 3 + /3 3 ) PQ + (a 2 P - /3 2 Q)Z(Pn- Qm) + a/3 {-P 3 -Q 3 -Z(mP 2 + wQ 2 )} = 0;