An introduction to the mathematical theory of attraction ...

Examples. 277
where
Jk k
sih a = -, sinh = -, 7 = a + j3,
c2 + a2 - b2 C a2 - b2
a cosh a = V(a2 k) = --- - = - + -
C2 +- b2 - a2 c b2 - a2
b cosh 3 = V/(b2 + k) = -c
2c 2 2c
We have then
dk ab cosh a cosh f da b cosh f d1B a cosh a dy 1
dc ck ' dc ck ' dc ck ' dc k'
whence
1 dq,1 ab coshcoshosh /3 % 1 C o b cosh 3 + ne cosh (a + n^y)
2 dc 2ck O sinh (a + n'y) 2c sinh2 (a+n'y)'
dq2i ab cosh a cosh I3 c 1 +<~ cosh n-y
dc ck i sinh n y i sinh2 ny'
1 dq22 ab cosh a cosh o 1 c ao cosh a + nC cosh (O + sy)
dc 2ck o sinh(/+ ny) - 2c sinh2 (B + n7)
6. From the values of q1î, q12, and q22, given in Ex. 1, Art. 126, find the
leading terms in the expression for the mutual force F between two charged
spheres.
d 12 + -- V2 2 y2P
del d l2 de2
- 1 - a2bc a3b2c(22 - 2b2 a2) }
(C2 ~ + (c2 _ b2h + ac)2(0c - b2 - ac)2
ab + b22(3c2-a2 - b2) ~3b32 {(c - e2- b2)(5c2-a2 - b2)- ab2}
2+ 12 i (c2 - 2 - b2)2 c2(c - a2 ab)2 (c2 - a 2 - b- ab)2
2- i 2 a2c 2b3C(22 - 2a2 b2)
- (c2 _ a2)2 ( a _ a2 + bc)2 (c2 _ a - bc)2
7. Show how to exhibit the series expressing the coefficients of capacity
and induction of two electrified spheres as rational functions of their iadii and
the distance between their centres.