398 M. G. Kirchhoff on the Motion of Electricity in Wires. 

 the equation 



From this expression for the density J, I deduce aii expression 

 for the strength i of the current, by multiplying the above with 

 p dp dyjr, and integrating the expression with reference to p from 

 to a, and in reference to yjr from to 2^ ; as V is indepen- 

 dent of p and yfr, when I make 



7^" Jo Jo 



WjO dp d-slr, 



■"^^ Jo Jo 



I obtain 



We have here 



cos^cos^' + 2i(log2r— 1) 



r.,ds' 



J "- 



jM M 3'p'dp'd'^'pdpdflog\/p^ + p'^~2pp'cos{ylr'-f). 

 Jo Jo Jo Jo 



2_ 



7ra' 

 The integral 



r77 



d^fr log ^p2 ^ pf2 _ 2pp' cos (>/r' — -f) 



is of the same form as that already considered and denoted by U : 

 from the conclusions there stated, it follows that the integral is 

 = 27rlogp', when p' > p, and =27rlog|0 when p' '<p- Mul- 

 tiplied by pdp, and integrated from to a, it therefore gives 

 this expression : 



^,.(loga_^'). 

 As we may set 



''a f^n 



3'p'dp'dyfr'=i, 



the third member in the expression for w will be 



= -2nog«+| "—^Vp'dp'd^ir'; 



Jo Jo "■ 



and hence we obtain 



m; = G ^ cos Q cos & + 2e (log?^ _ 1 ) + Pr' '^-J^^'p' dp' df. 



The remaining double integral cannot be reduced to a simple 

 form, as J' is an unknown function of p' ; its value, however, can 



