where the characteristic k(o) is, say, continuous on the unit sphere and its integral vanishes there. In this paper I shall consider the operation (1) as the convolution of f with a distribution K in the sense of Laurent Schwartz, as it has been done already for n=1 by Schwartz himself [17, p. 115]. This point of view permits one to disregard the delicate question of the existence of the limit in (1) and to apply 3C more generally to a distribution T rather than to a function f. We shall see that K belongs to one of the spaces where the Fourier transform !Y(K) is defined. This gives an immediate solution, at least theoretically, to the problem of composition of singular operators: If 3c1 and 5C2 are defined by the distributions K1 and K2 respectively, then the distribution K which corresponds to X1I 3X2 will have 3Y(K) = Y(K1) f(K2) as its Fourier transform. The expression Y(K) (which is in reality a function) has been introduced by Mihlin [151 and Giraud [9] under the name of 'symbol of the operator 3e. As an application I shall calculate 5(K) in the special case when k(a) is a homogeneous harmonic polynomial. The corresponding result has been announced first by Giraud [91 and a proof has been given more recently by Bochner [1]. The present method uses an apparently new process of generating a complete system of spherical harmonics with the aid of Grassmann's algebra with complex multiplication which is described in ?3 and can be read independently of the rest of the paper and of the theory of distributions. On the other hand in ??2 and 4 I make constant use of the theory of distributions and conserve the notations and terminology of Schwartz's book [16; 17], except that I note an integral extended over Rn by only one integral sign. In a paper, now under preparation, I shall reconsider from the present point of view the known results concerning the composition and inversion of singular operators [9; 10; 15; 4, footnote p. 261].