It is shown that by using the methods developed in papers I–III of the present series it is possible to reduce the problem of deriving the solution of a certain class of dual relations involving Jacobi polynomials to that of solving an integral equation of Schlömilch kind.

A study is made of the generalization of the entropic law (xy)(zw)—(xz)(yw) to an identity connecting two operations. It is shown that such an identity is equivalent “in the large” to the condition that the set of endomorphisms with respect to one operation is closed with respect to the other. Furthermore, for such entropic operations, each may be regarded as a generalized endomorphism of the other and various generalizations of elementary properties of endomorphisms are obtained. Examples of entropic pairs of operations are quite common in mathematics and a number of these are discussed. An important aspect of the paper is the matrix notationintroduced to facilitate what would otherwise be extremely cumbersome computations with entropic operations.

It is generally agreed that the long-range alpha particles of fission are set free before the fragment nuclei have acquired more than a small fraction of their final energy of separation, but whether the alpha particle is liberated before the instant of scission, at that instant, or from one of the fragment nuclei very shortly thereafter, has remained an open question. Each of these views has been seriously advocated. These various hypotheses are examined in relation to recently published information regarding the distribution of mass in low-eneigy ternary fission, and other considerations, and it is suggested that the hypothesis having the strongest claim to attention is that which assumes that the alpha particles originate in the heavy fragments exclusively, being liberated, very shortly after the instant of scission, with probability not much less than unity, from fragment nuclei of low yield and small neutron excess. Conclusions which would follow, if this hypothesis were accepted, are indicated, and possible experimental tests of these conclusions are suggested.