is known, then clearly x0x1 + x1x2 + x2x3 + x3x4 + x4x0 can be determined by the solution of a quadratic equation. Moreover, the sum and product (x0 + [epsilon]x1 + [epsilon]^2x2 + [epsilon]^3x3 + [epsilon]^4x4)^5 and (x0 + [epsilon]^4x1+[epsilon]^3x2 + [epsilon]^2x3 + [epsilon]x4)^5 can be expressed rationally in terms of x0x1 + x1x2 + x2x3 + x3x4 + x4x0, [epsilon], and the symmetric functions; [epsilon] being a fifth root of unity. Hence (x0 + [epsilon]x1 + [epsilon]^2x2 + [epsilon]^3x3 + [epsilon]^4X4)^5 can be determined by the solution of a quadratic equation. The roots of the original equation are then finally determined by the extraction of a fifth root. The problem of reducing an equation of the fifth degree, when not soluble by radicals, to a normal form, forms the subject of Klein's _Vorlesungen uber das Ikosaeder_. Another application of groups of finite order is to the theory of linear differential equations whose integrals are algebraic functions. It has been already seen, in the discussion of discontinuous groups in general, that the groups of such equations must be groups of finite order. To every group of finite order which can be represented as an irreducible group of linear substitutions on n variables will correspond a class of irreducible linear differential equations of the nth order whose integrals are algebraic. The complete determination of the class of linear differential equations of the second order with all their integrals algebraic, whose group has the greatest possible order, viz. 120, has been carried out by Klein.