The present chapter is in the nature of a Note, collecting together briefly details in regard to the characters of the various kinds of algebraic integrals which present themselves for a fundamental algebraic equation f(x, y) = 0.

From the point of view in which the equation f(x, y) = 0 is regarded as representing a plane curve, in the case in which this curve has no multiple points other than double points with distinct tangents, and the lines x = 0, y = 0 are in general positions with respect to this curve, we have shewn that, if p be the number of everywhere finite algebraic integrals belonging to the curve, then p is also the number of adjoint polynomials of order m – 3 (if m be the order of the curve); and further that these polynomials have each 2p – 2 zeros on the curve (other than at the multiple points), no one of which is common to all of them. Also that the number of tangents which can be drawn to the curve from an arbitrary point is 2m + 2p – 2. We have also shewn that the theory, for a curve whose multiple points have any complexity, can be reduced to this case, by birational transformation, which will leave the value of p, as defined, unaffected. We have then shewn, for the simple case, by means of the formula w = 2m + 2p – 2, using loops of integration, that a general algebraic integral without logarithmic infinities, associated with the curve, has 2p linearly independent periods; and we have obtained this same result by use of the Riemann surface.