1522 
‘ 1 
LG aa ch eee 
It will be seen from the meaning of fand g that these functions will 
not depend upon the direction of the radius-vector 7 = Va? ty 2. 
It gives, however, no simplification to introduce this in (1). Now 
develop g under the integral sign, according to powers of z, y and z. 
The differential quotients of gy can then be brought before the integral- 
sien, and there remain integrals of the general form 
Epos 
{hy arysz'f (a,y,2) dadydz 
As f only depends upon r, these integrals will vanish when one 
or more of the numbers 7, s, ¢ are odd; and their value will remain 
unchanged by transposition of 7,s and 7. If we only go to terms of 
the second order, besides the value # for 7 —s—=t=O, we also 
meet the integral with 7 = 2 s=t=0, which has already been 
represented Le. by’ €. (The function f will only differ from zero 
for the small value of 7, for which the ‘molecular attraction is yet 
sensible. Therefore we could call the magnitude W/8.e, in analogy 
with the “mean error’, the mean radius of the sphere of attraction). 
The just mentioned manipulation makes the integral-equation pass 
into a differential equation for g: 
ee EC . 
9 (#54121) — Fg 9 (<4 ar dy” SSF el = Gh Cho Za) 
or, when we now introduce 7 
gee 
CSE clas Mie ds Ricca eine ANN 
dy? | r dr & & 
The general solution of the equation (8) without second member is 
XT 
A 
Arte + Brie 
OE) 
-—— and from this we easily find by variation 
in which x? = 
of constants the solution of (3). The two constants in this solution 
may be determined from the two following conditions. In the first 
place g must remain finite for =o, a condition which was already 
required with the integral-equation (1). If we take for the second 
condition, that gy remains finite also for = 0, then we shall find 
2 sinh xr xs 
a7 De deren) 
ear 
XT 
9 7 
2e : ‘ 
g = —— Js (s) sinh xs ds 
Eer , 
) 
