XXVII. On curves of the third order

The following Notes are intended as supplementary to Mr. Cayley's Memoir on Curves of the Third Order. If in a cubic U we substitute x + λx', y + λy', z + λz' for x', y', z' let the result be written U + 3λS + 3λ²P + λ³U', where S and P are evidently the polar conic and polar line of x'y'z' with respect to the cubic, and 3S = x' dU/dx + y' dU/dy + z' dU/dz : 3P = x (dU/dx)' + y (dU/dy)' + z (dU/dz)'.