 Mathematical functions are not mentioned by Russell in the 1924 list of “logical constructions” although the analysis of mathematical functions is the principal application of the theory of definite descriptions in PM. The basic “functions” of PM are propositional functions. The Greek letters \(\phi, \psi, \theta, \ldots\) are variables for propositional functions, and, with individual variables \(x, y, z, \ldots\) go together to form open sentences \(\phi(x), \psi(x,y)\), etc. This is the familiar syntax of modern predicate logic. Mathematical functions, such as the sine function and addition, are represented as  term forming operators such as \(\sin x\), or \(x + y\). In contemporary logic they are symbolized by function letters that are followed by the appropriate number of arguments, \(f(x),g(x,y)\), etc. In chapter *30 Whitehead and Russell propose a direct interpretation of such expressions for mathematical functions in terms of definite descriptions, which they call “descriptive functions”. Consider the relation between a number and its sine, the relation which obtains between \(x\) and \(y\) when \(y = \sin x\). Call this relation “\(\text{Sine}(x,y)\)” or more simply, “\(\bS(x,y)\)”, as a two-place relation. The mathematical function can then be expressed with a definite description, interpreting our expression “the sine of \(x\)” not as “\(\sin(x)\)”, but literally as “the Sine of \(x\)”, with a definite description, or “the \(y\) such that \(\text{Sine}(x,y)\)”. Using the notation of the theory of definite descriptions, this is ‘\((\iota x)\bS(x,y)\)’. The effect of this analysis is that Whitehead and Russell can replace all expressions for mathematical functions with definite descriptions based on relations. This definition involves relations in extension, which are represented with upper case Roman letters and with the relation symbol between the variables. The definition in PM is: *30.01. \(R`y = (\iota x)xRy\), with the notation \(R`y\) to be read as “the \(R\) of \(y\).” As with the theory of descriptions, the result of this definition is to facilitate the proofs of theorems which capture the logical properties of mathematical functions that will be needed in the further work of PM.