Gamma Function to an Electrostatic Problem. 255 

 dlovTin) rdfocrT(n) 



dn 



L dn Jn=:i 



The quantity — — J constitutes " Euler's constant," 



but neither it, nor the values of the differential logarithm cor- 

 responding to any other value of n, can be presented in algebraic 

 terms. 



Now let H(tz), which we may call the Eta Function, stand 

 for 



f- (/logI» -j _ dhgr(n) 

 L dn J n=1 dn 



so that by (2) 



-H W =( 1 -J) + (J- ? i T ) + fi -_i,) T ....: (3) 



It is obvious from this that H (0) = oo , and H(l) = ; also that 



for values of n between and 1, R(n) is positive and decreases 



as n increases, while when n>l it is negative. 



b "'"' 



Reverting now to (1), put «= r , then 



° l a + b 



Q a a% 2 «=* 1 



Y (« + 6) 2 S=l 5(5 — 1+71) 



__ a% 1 ^"/l _1 \ 



{a + b) 2 ' n-1 s=l \s 5-l + ?i/ 



— jLJ'i 1 -*! L + LJ_+ I 



a + b\ n^2 n+l."*" 3 ■ n + 2 ' j 



l. £. 



~H(n)by(3); 



Q« ab 



This gives the charge on the sphere of radius a in terms of 

 an Eta function. Similarly the charge on the sphere of radius 

 b is given by 



Qf = J* H(-ij) . . . . (5) 



V a + b \a + bJ v ' 



