Quine observes that Russell's introduction of the notion of "apparent variable" had the following result:

Definition and the real variable and 16 Propositions connecting real and apparent variables was abandoned in the second edition.

"Let us give the name of matrix to any function, of however many variables, which does not involve any apparent variables.

A function involving a first-order function or proposition as apparent variable will be called a second-order function, and so on.

An introduction to the notation of "Section B Theory of Apparent Variables" (formulas 8- 14.34)

This new section eliminates the first edition's distinction between real and apparent variables, and it eliminates "the primitive idea 'assertion of a propositional function'.

For example, if one asserts that " y: f(x, y) is true", then x is the apparent variable because it is unspecified.

At the outset of his Introduction he declares "there can be no doubt ... that there is no need of the distinction between real and apparent variables...".

By all available measures, business has increased significantly for 1994, improving the company's estimated 50 percent share of the weight-control market, and the only apparent variable has been Ms. Sullivan.

In *12 of Principia Mathematica (1913) he defines "a matrix" as "any function, of however many variables, which does not involve any apparent variables.

It says that any function is coextensive with what he calls a predicative function: a function in which the types of apparent variables run no higher than the types of the arguments".

It will be observed that, in a hierarchy in which all the variables are individuals or matrices, a matrix is the same thing as an elementary function [cf 1913:127, meaning: the function contains no apparent variables].

Quine explains the ramified theory as follows: "It has been so called because the type of a function depends both on the types of its arguments and on the types of the apparent variables contained in it (or in its expresion), in case these exceed the types of the arguments".

Second edition, abridged to *56, with Introduction to the Second Edition pages Xiii-xlvi, and new Appendix A (*8 Propositions Containing Apparent Variables) to replace *9 Theory of Apparent Variables, and Appendix C Truth-Functions and Others.