Mr. A. Cayley on Curves of the Third Order. 67 
of which the complete integral may be expressed in the form 
d n—i j ; 
(sin @)"(sin 0 70 sin ) (sin 0)"(C, + C, | da(sin a)-"), 
at least in the case in which 7 is an integer not greater than u, for 
which case this form is here demonstrated.) 
If it be assumed that the solution of (2), obtained on the suppo- 
sition that is an integer, may be extended to the case in which z is 
a general symbol, it follows that the solution of (1) will be obtained 
from it by changing 2 into rz. This would give 
a 
d d 
MeN rye ae "Or Spat: 
u=(sin0) (sin 6 sin ) {7(,, e? =! tan 3) 
+F(», e-¢%-1 tan 5) \, 
which is easily shown to be equivalent to 
beter) {ie ovn1 6 ony re al aS 
u=f\| psin O76 sin 6, e tan 3 +F(o sin @ qn, e¢ tan5 > 
where p=7(sin @)~’, but p is to be treated as a constant till after all 
operations. 
This expression is shown to give known particular integrals, such 
as (1—2r cos 0+7")~#, and 
cee) eee Be s hb etl 
7 (sin 0) (sin 076 sin 0) (tan 5) COS 7. 
It appears probable, therefore, that the generalization of the result 
obtained for the limited value of x is legitimate; but the author does 
not profess to demonstrate this conclusion, believing that the principle 
of the “‘ permanence of equivalent forms’? is not at present established 
in such a sense as to amount to a demonstration. 
“ A Memoir on Curves of the Third Order.” By Arthur Cayley, 
Esq., F.R.S. 
A curve of the third order, or cubic curve, is the locus represented 
by an equation such as U=(x {2, y, z)°=0; and it appears by my 
* Third Memoir on Quantics,” that it is proper to consider, in con- 
nexion with the curve of the third order, U=0, and its Hessian 
HU=0 (which is also a curve of the third order), two curves 
of the third class, viz. the curves represented by the equations 
PU=0 and QU=0. ‘These equations, I say, represent curves of 
the third class; in fact, PU and QU are contravariants of U, and 
therefore, when the variables w, y, z of U are considered as point 
coordinates, the variables £, n, of PU, QU must be considered as 
line coordinates, and the curves will be curves of the third class. I 
propose (in analogy with the form of the word Hessian) to call the 
two curves in question the Pippian and Quippian respectively. A 
eometrical definition of the Pippian was readily found; the curve 
is in fact Steiner’s curve Ry mentioned in the memoir “ Allgemeine 
