CONSTITXTTION OP MATTEE AND AXALTTICAL THEORIES OF HEAT. 



49 



the first derivative, on the right, of qofi) at and (p'{i^ — 0), the derivative 

 on the left, is similarly defined. When g^'d^ + O) and 9' (1^ — 0) are equal, their 

 common value , the first diiferential coefficient of (p (|) at , is represented by 



Let X be the continuous variable which has for its domain the continuum 

 in which the domain of | lies ; also let x^, x^ be any two values of x, x^ being 



greater than x^. Then I will call ] (p (|) di, the integral of (p (|) between the 



'x, 



limits x^ and x^. For the sake of clearness I proeeed to State the exact meaning 

 of this Symbol. 



Divide the interval {x^, into n partial intervals d^, 8,, ... 8^, ... ö^ . Let 

 and be respectively the upper and lower limits of (p (|) in the interval 8^ ; 



also let and stand for 2 K 2 ^» respectively. Then, if 6', and 



1 I 



converge to one and the same limit when n approaches infinity and 8^ approaches 



zero, this limit is the quantity represented by / q){^)di,; and 93 (^) is said to 



be integrable in the interval {x^, x^). 



It should be noted that, contrary to what holds true in continuous analysis, 

 the continuity of (p{^) at every point in the domain of | does not involve its 

 uniform continuity in that domain. I will call ^(l) a continuous function of § 

 when it is uniformly continuous in the domain of |. Thus, when cp(^) is a con- 

 tinuous function of | it is integrable. More generally, if there exists a function 

 f{x), integrable in x and, further, such that /"(§) = (pi^), then cp{^) is integrable. 

 For the sake of brevity I will call f (x) an associate function to (p (5). It should 

 be noted that if |/'(^)|^_| exists it equals 9>'(|). 



Formulation of an Improperly continuous theory of solids. 



45. By an ether theory of matter is understood a theory which , while 

 looking upon ether as a continuous medium endowed with perfect fluidity, re- 

 gards matter as ether in motion^). If, therefore, an ether theory be thorough- 

 going , i. e. , if it does not stop short at certain finite bodies , called atoms or 

 molecules, simply regarding them as ethereal structures whose internal Constitu- 

 tion is inscrutable ^) , but professes to know all about the internal Constitution 

 of any material body however small ; then it is possible to exactly describe the 



1) See the article „Matter as Non-Matter in Motion" in Pearson's book, Chapter VII. 



2) Cf. L armor, 1. c. p. 455. It should be noted that those ether theories of matter which 

 are not thorough-going — for example, the vortex-atom theory of Lord Kelvin and the theories of 

 H. A. Lorentz, Wiechert, and Larmor — belong to the type discussed in Part II. 



Athauaig. d. K. das. d. Wiss. zn Göttingen. Math.-Phys. Kl. N. i\ Bp-nd 2,4. _ 7 



