Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 343
Surface 01 02 603 4 Reference letter
2y2 + 2y (z, w)2 + (z, W)4= x -Y O O A
z2 (, y) + - (x, yy)22 w2 (, + )32 =0 y -z -W B
(XW - yZ)2 + 2 (xw - yZ) (z, w)2 + (z, w)4 = z w O O C
(2xw - y2)2 2 (2xw - y2)(z, )2+ (z, w)4 = O w O O D
ax2y2 + bz2w2 + cxyzw + dxw3 + eyz3 = 0 3 -3y z -w E
Of these surfaces (A), (B) are the two which Poincaré discovered*; the existence
of integrals of the first kind on the other three surfaces was pointed out by the author
in a note published in the Conptes Rendus for 2 Sept. 1899.
~ 5. BIRATIONAL TRANSFORMATIONS OF THE SURFACES INTO CONES.
Corresponding to each of these surfaces we can at once construct a total differential
by the formula already given (~ 1); and since the 0's are unique, there cannot be on
each surface more than one such differential, the integral of which remains finite. But
it has not been proved that such an integral does actually remain finite everywhere.
This could be done by examining its behaviour at each singularity of the surface. But
it is of interest to shew that each surface can be birationally transformed into a nonsingular cubic cone, and as such a cone admits necessarily of an integral of the first
kind, we are thus incidentally assured of the finiteness of our integrals. The transformations which effect this object are as follows:
(A) We write the equation in the form
fxy + (Z, W)2}2 = {(Z, W)22 - (Z, w)4,
and choose as a new variable w one of the factors of the right-hand side, so that the
latter becomes w (z, w)3.
The quadric transformation:: y: z: w =x'w'-(z', W'): )y: y'z' y 'w)
x ' y': z: w' = y + (, w)2: yw: zw: w2 w
* Loc. cit.