56 The Law of Distribution [CH. iv 



The product of expressions (114) and (115) gives v p . From this and equation 

 (113) we get 



...dx b dx c ...dy b dy c ... 



, v, w)auavawaxayaz-rr77T~- ...(116). 



... dx a dx b dx c ... dy a dy b dy c 



64. Let us define condition q as the condition that there shall be a 

 molecule having its coordinates within the limits x' and x' + dx', etc. The 

 volume v pq for which conditions p and q are both satisfied will consist of 

 contributions from different pairs of molecules. In expression (116) we 

 suppose molecule A to satisfy condition p. If molecule B satisfies condition 

 q the corresponding contribution to v pq /v can be obtained from the right-hand 

 of (116) by limiting the integration in the numerator to the range from x' to 

 x' + dx as regards x b) and to similar ranges as regards y by z b . The number 

 of molecules capable of taking the rdle of B is 



Nf(u, v', w')du'dv'dw'. 



Hence we obtain as the value of Vv 



u, v, w)f(u, v', w')dudvdwdu!dv'dw'dxdydzdx'dy'dz' 



1 1 . . . dx c . . . dy e . . . 

 . . . dx a dx b dx c . . . dy a dy b dy e 



...... (117). 



The integration extends throughout all the values of the variables except 

 such as are excluded by the conditions of 33. In applying these conditions 

 to the numerator, we must replace x a , y a , z a by x, y, z and x b , y b , z b by x', y', z'. 

 We therefore find, as we ought, that v pq vanishes when the points x, y, z and 

 x' ', y', z' are at a shorter distance than cr, or when either of them is at a 

 distance from the boundary less than ^a-. We also see that v pq /v is not equal 

 to the product of v p fv and v q /v, so that the fulfilment of conditions p and q 

 cannot be treated as independent events. 



65. In the special case in which the radii of the molecules are vanish- 

 ingly small, those parts of the generalised space which are excluded in 33 

 may be neglected. In the integrals of equations (116) and (117) the integra- 

 tions with respect to the variables with different suffixes now become inde- 

 pendent. We may for instance write 



JjJJjJ ... dx a dx b dx c ... dy a dy b dy c ... = ljl dx a dy a dz a \\\ dx b dy b dz b ... = fl N . 



The other integrals can be simplified in a similar manner, and we obtain 



v N 



= 7=r/(w, v, w)dudvdwdxdydz 



v /-Zy\ 2 



-? = / - J f(u, v, w)f(u', v', w')dudvdwdu'dv'dw'dxdydzdx'dy'dz'...(~\.l$}. 



