116 Prof. R. Clausius on the Theorem of the Mean Erg ah 



Further, of the six boundary-planes, only those two come into 

 consideration which are perpendicular to the ^'-direction. Lastly, 

 we will, as before, denote the mean value of that part of the ergal 

 which relates to the ^-direction by mk, and also introduce the 

 symbols g and g ] 3 their significations being determined by the 

 equations 



d\l , d\J /n 7 \ 



™0=^-> ™J=2j < 87 ) 



In employing these symbols mean be omitted from all the terms, 

 and the equation in question becomes 



8h = w8 log u-\-g8c- i rg'8c } i .... (88) 

 or, replacing w by Xtiz*, 



8h = Wz*8\ogu+g8c+g'8c ! . . . . (89) 



This equation, in which any definite value is assumed for the 

 quantity z, can, neglecting infinitesimal differences, be applied 

 to all the points with which this quantity lies between the values 

 z and z + dz, and the number of which is ^~N/(z)dz; we here 

 presuppose that the function /has the same form in the changed 

 as in the original motion. The equation referred to these points 

 may now be multiplied hy f(z)dz and integrated from £=0 to 

 a: = cc ; and it is to be remarked that the quantity z, which in 

 the changed motion has the same infinite series of values as in 

 the initial motion, with the variation indicated by the symbol 8 

 is to be regarded as constant ; and hence the factors z 2 and f(z)dz 

 may at pleasure be put before or after the variation-symbol, and 

 in like manner the symbols of integration and variation may be 

 interchanged. The. new equation may therefore be written 



8 \ <X> hf{z)dz=tv8 f "log u . z*f{z)dz + (™gf(z)dz . 8c 



Jo Jo c/0 



+rv/w*.w (90) 



The integrals 



Chf{z)dz, ("jtfW* and ("*//(*)&. 



Jo Jo Jo 



according to what has been said above, denote the arithmetic 

 means of all occurring values of h, g, and g 1 , which means we 

 will designate by h, g } and g'. For a simpler notation of the 

 fourth integral we will use the letter it, the meaning of which is 

 determined by the equation 



log u.z 9 f{z)dz (91) 



logU= 1 

 Jo 



