244 PROCEEDINGS OF SECTION A. 



was necessary to draw roughly the equipotential lines in the 

 interior of coils of different lengths, and the curves so obtained 

 proved interesting enough to indiice him to go into the calculations 

 more fully and attempt to draw the curves more accurately, both 

 inside and outside the solenoid. The equipotential lines have 

 therefore been drawn for four coils, whose lengths are — (1), L = a; 

 (2), L = 2a; (3), L = 4«; (4), L = co (Plates II., III., lY., 

 and Y.), where L is the length of the solenoid and a the radius of 

 its transverse section. These coils will be referred to as coils 

 (1), (2), (3), or (4). The formulae used are given in "Maxwell's 

 Electricity and Magnetism," vol. ii., p. 284, second edition, and 

 are as follows : — 



The magnetic potential at any point outside is 



n = ny(Y, - YO; 



at any point inside it is 



O = ?» J (— 4 TT z + Yi — Yo), 

 where 



O = magnetic potential at the point considered : 



n =■ number of turns of wire per unit of length : 



J = current in C.G.S. imits : 



Yi = potential at the point due to a plane area of surface density 

 unity at the positive end of the solenoid : 



Ya = potential at the negative end : 



z = distance of the point from the centre of the solenoid measured 

 along the axis : 



the values of Yi and Yo are found from the expressions 



Y = 2 ^ i — rPi + a + ^ -Po — — C, P4 + ,&c.,whenr< a, 

 { 2 a 2-4 a 



\ = 2 TT { — ? P2 + r P4 — , &c., when r > a. 



For points along the axis the simpler formida Y= 2 tt (i/a^ + ^^ 

 — rj was used, where 



r = distance of the point from the centre of one of the circular 

 ends of the solenoid. 



Pi, Po, &c., are the zonal surface harmonies of orders 1. 2, &c., 

 corresponding to the angle which r makes with the axis of the 

 coil. 



The values of Y^ and Yo at different points inside and outside 

 ihe solenoid were worked out by means of these formulae, and the 

 cbrrespondmg values of Q were thus found for between forty and 



