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1. If a hyperelliptic integral of genus 2 is reducible to an elliptic integral by a transformation of order k then there always exists a second integral t with the same irrationality which is reducible by a transformation of the same order. In the algebraic treatment of the problem the difficulty lies in the determination of the second integral after the first has been constructed. For k = 4 the problem has been treated by Professor BOLZA : and in the sixth section of his dissertation the formulhe for the two reducible integrals are established algebraically. Professor BOLZA used a particular system of variables x1 x2. These are chosen so that if

transactions of the american mathematical society

On the system of a binary cubic and quadratic and the reduction of hyperelliptic integrals of genus two to elliptic integrals by a transformation of the fourth order