492 DR. A. N. WHITEHEAD ON 



itself to rewriting with appropriate changes a chapter of any modern treatise of 

 electricity and magnetism. It would seem necessary to subdivide the class of 

 particles into " positive " and " negative " particles, a charged volume containing an 

 excess of one type. The conception of an ether conveying lines of force is replaced 

 by the class of the linear objective reals. The details can be managed much as in the 

 analogous case of Concept V., considered later. An indefinite number of extraneous 

 relations are required to " locate " the particles, just as in Concept I. This concept 

 (as thus developed with "particles") is not completely a "linear" concept. It is a 

 hybrid between the "linear" and "punctual" concepts. In its dualism it is not 

 superior to the classical concept. But, in possessing moving linear objective reals as 

 well as moving particles, it is richer in physical ideas. 



Concept TVs. In this concept, just as in Concept II., each triadic extraneous 

 relation of Concept IVA. between an instant of time, a particle, and a point is 

 replaced by a dyadic extraneous relation between a point and an instant of time. 



PART IV. (i) THE THEORY OF DIMENSIONS. 



*3. Concept V. depends upon a treatment of the theory of dimensions different 

 from -that which at present obtains. The theory here developed is relevant to any 

 definite property which (1) is a property of classes only, and (2) is only a property of 

 some classes. It will be clearer, and no longer, to explain the theory in its full 

 generality, and in Concept V. to make the special application required. 



This general theory of dimensions may, perhaps, have a range of importance 

 greater than that which is assigned to it in the sequel. In *10 a set of hypotheses 

 are given respecting the property ^> ; and when these are true of <, the propositions 

 and definitions of *3 to *8 acquire importance and emerge from triviality, also in this 

 case further deductions of propositions can be made. The Concept V. to which this 

 theory is applied is explained in the definitions of *20 and the axioms of *22. In 

 this Concept V. a special property <j> is taken, which is termed "Homaloty" (cf. 

 *20'11'12), and (cf. *22) in the axioms a relation R is considered such that 

 " homaloty," defined in respect to R, has the properties of the axioms in *10. 



*3'01. Definition. If <^\x is some proposition involving the entity x, which may 

 be varied, so that (j)lx and <j>ly make the same statement (<) about x and y 

 respectively, then any entity z, for which (f> I z is true, is said to possess the property <j). 



*3'02. Definition. A $-class is a class with the property <, that is to say, if u is 

 a <-class, then <j)lu is true. 



*3'11. Definition. The ^-region is the logical sum of all classes which possess the 

 property <. The symbol O^ will denote the (^-region. The symbolic definition is 



Df 



*3'12. The common (f>-subregion for u is that class which is the common subclass 



