352 Dr. E. B. Rosa on the Specific Inductive 



are nearer together than the axes of the cylinders. The 

 potential at any point P distant R' and R" from these lines is 



V = logR'-logR". 



Expanding log R' and log R" in terms of R, r, and the 

 angle <£, we have approximately 



cos <p ^ + - cos 2</>jr- 2 + 5 cos 2><f> ^3 4- J 



r r 1 ?- 2 1 r 3 1 



log R"=log R- [cos </>i g+ 2 C0S 2( ^R2 + 3 cos3 ^ii3 + • • -J 



(j) = ot—0 and <f) l = ot + ; hence 



R 3 

 + gsin3«sin3^ + ...].(C) 



Inside the cylinder the potential is 



V'^aj r sin + a. 2 r 2 sin 2d + a s r* sin 30 + 



Outside the potential is 



V" = V + ^ sin + % sin 20 + h -l sin 30 + ... . 



V being the potential when the cylinder is absent, and 

 «i? a 2i a 3 ' • • an d &i> ^2) h • • • being undetermined coeffi- 

 cients. Substituting the value of V givm by equation (C) 

 in the expression for V", we have 



V"= [--2 ^sin a] sin 0+ \k - | ^ sin 2«"|sin 20 



log R'-log R"=V = -2 [sin a sin ~ + Lin 2a sin 1 



+ [^~3'Er 3Sin3a J sin8<9 



At the surface, where r=p the radius of the cylinder, the 



conditions are that V'=V" and K'^=K"^ r . These 



equations determine the coefficients a^ a 2y a 3 . . . b u b 2 , b 3 . . . 

 and give 



V" = 2{(K^-r)^sin^ + i(Kg-^) S -^ S in2^ + 

 J(K^)^i.M + ..:}, (D) 



where K=tT7 — j^,,. Differentiating this value of V", we 

 have, after putting r = p, 



