Theory of Magnetism. 49 



gous resolved velocities due to the moveable needle. Then, p' 



and V being the density and velocity at that position, by what 



is shown above 



_Y1 / V' 2 \ 

 pt=p e **- Po \l-— J nearly, 



and 



1 -^ = 9^{( W l +W 2) 2 +(^+^) 2 +K + ^ 2 ) 2 } 



Now the velocity and density being functions of space only, it 

 is easy to see that the accelerative action on any atom must have 

 a constant ratio to the acceleration of the fluid where the atom 

 is situated. I have found that this ratio is independent of the 

 magnitude of the atom (Principles of Mathematics, p. 315) . As 

 the moveable needle is capable of motion only about a vertical 

 axis through its middle point, we are concerned exclusively with 



a force proportional to j-, y being the distance from the 



axis. The stream of the fixed needle is symmetrical with respect 

 to a vertical plane through its axis, and flows nearly perpendi- 

 cularly to the axis of the moveable needle, so that u x is very small 

 at the positions of all its atoms. A little consideration of the 

 courses of the streams will suffice for perceiving that neither the 



forces proportional to [u x + u 2 ) ( -j+ + ~ ), nor those proportional 



> , , ,(dw x , dw^\ Y V y , * w , 



to {w x + w 2 ) I -~ + -7-fh produce any momentum ot rotation 01 



the needle. Consequently the motion of rotation wholly depends 

 on the forces proportional to 



, x /tfo, dv 2 \ 



dy dy 



dv 

 Now the forces v x -— evidently produce equal and opposite 



momenta on the north and south arms of the needle; the same 



dv 

 is the case with the forces v 9 -j- 2 , because the values of v are 



d y . . . 



equal with opposite signs at equal distances on the opposite sides 



dv 

 of the centre of motion. Also the forces v 2 ■— are mutually de- 

 structive, because v 2 at any distance from the centre of motion 

 has equal positive and negative values on the opposite sides of 

 the axis. There remains, therefore, only the momentum due to 



the forces v x j- 2 . These will clearly tend to produce rotation, 

 Phil. Mag. S. 4. Vol. 38. No. 252. July 1869. E 



