218 Prof. G. M. Minchin on the Magnetic 



when added to (59) give as the equation of a line of force 

 near the wire, 



L cos d> ( , T * x c 2> ) lfr /3c 2 m 2 \ 



+ T- 1 f ! - cos2 *( L T + I- w2 -8&)}= const - ( 7 °) 



As before, we can verify this result by showing that the curve 



(70) is at right angles to the curve (48 *). Taking, as pre- 



c 2 

 viously, j — g = A,, this equation is 



m 



(L-l-X) cos 0+^-5 !(3\-J)L + i-15\ 



— (iL — 1 +\-iX*) cos 20}= const. (71) 



It is interesting to observe that the supposition of variable 

 density affects the term of the first order both in the value of 

 <t) and in the constant of the line of force in the same way. 



To trace any Line of Force. — With the centre. D, of the 

 cross-section of the wire describe a series of circles, fig. 3, 

 their radii being D/, T>g, J)h, . . . Then to trace the par- 

 ticular line of force w-hich passes through f (suppose) let 

 D/=m , and let m be the radius of any other of the circles. 

 If cf> is the angle defining the point in which this latter circle 

 is met by the line of force through /, w r e equate the left-hand 

 side of (60) in case of constant density, and of (71) in case 

 of variable density, to the expression which this left-hand 

 side assumes when m and are put for m and </>. Firstly, 

 neglecting the terms of the second order, we have, to deter- 

 mine 0, in the case of constant density, 



iL-^(L-l + \)cos<£=|L -^(L -l+X ); 



and if e is the value of given by this equation, we can put 



20 = 2e in the terms of the second order. If the term of the 



in 

 second order, ^- 2 {- • •}> m (60) is denoted by y, the more 



correct value of is obtained from the equation 



_(L-l+\)(cos0-cose) = y- 7o , 



where in y we put 2e for 20, and in y , of course, 20 = 0. 



As a numerical example, let the wire in fig. 1 form a circle 

 20 millim. in diameter, i.e. OD in fig. 2 is 10 millim.; and 

 let the diameter of its cross-section be 2 millim. i. e. DB = 

 1 millim. in fig. 2. 



