be a linear differential equation, the coefficients in which are rational functions of x, and let y1, y2, ..., y_n be a linearly independent set of integrals of the equation. In the neighbourhood of a finite value x0 of x, which is not a singularity of any of the coefficients in the equation, these integrals are ordinary power-series in x - x0. If the analytical continuations of y1, y2, ..., y_n be formed for any closed path starting from and returning to x0, the final values arrived at when x0 is again reached will be another set of linearly independent integrals. When the closed path contains no singular point of the coefficients of the differential equation, the new set of integrals is identical with the original set. If, however, the closed path encloses one or more singular points, this will not in general be the case. Let y'1, y'2, ..., y'_n be the new integrals arrived at. Since in the neighbourhood of x0 every integral can be represented linearly in terms of y1, y2, ..., y_n, there must be a system of equations y'1 = a11y1 + a12y2 + ... + a_(1n)y_n, y'2 = a21y1 + a22y2 + ... + a_(2n)y_n, . . . . . y'_n = a_(n1)y1 + a_(n2)y2 + ... + a_(nn)y_n, where the a's are constants, expressing the new integrals in terms of the original ones. To each closed path described by x0 there therefore corresponds a definite linear substitution performed on the y's. Further, if S1 and S2 are the substitutions that correspond to two closed paths L1 and L2, then to any closed path which can be continuously deformed, without crossing a singular point, into L1 followed by L2, there corresponds the substitution S1S2. Let L1, L2, ..., L_r be arbitrarily chosen closed paths starting from and returning to the same point, and each of them enclosing a single one of the (r) finite singular points of the equation. Every closed path in the plane can be formed by combinations of these r paths taken either in the positive or in the negative direction. Also a closed path which does not cut itself, and encloses all the r singular points within it, is equivalent to a path enclosing the point at infinity and no finite singular point. If S1, S2, S3, ..., S_r are the linear substitutions that correspond to these r paths, then the substitution corresponding to every possible path can be obtained by combination and repetition of these r substitutions, and they therefore generate a discontinuous group each of whose operations corresponds to a definite closed path. The group thus arrived at is called the group of the equation. For a given equation it is unique in type. In fact, the only effect of starting from another set of independent integrals is to transform every operation of the group by an arbitrary substitution, while choosing a different set of paths is equivalent to taking a new set of generating operations.