48 Prof. Challis on the Hydro dynamical 



p being the density where the fluid is undisturbed. So for an- 

 other set of streams 



p 2 -p e uf. 



But the steady motions to which these formulae apply may coexist, 

 (This proposition I have proved in the Philosophical Magazine 

 for February 1861, and in the ' Principles of Mathematics/ 

 p. 242.) Consequently the differential 



{u l + w 2 ) dec + (i?! + vS) dy + (w Y + w 2 ) dz 



applies to the steady motion compounded of the two sets, and is 

 plainly an exact differential. Hence if p' be the resulting den- 

 sity and V the resulting velocity, we have 



V'2 



p' = p e-^. 



Having determined the character of the magnetic streams of a 

 cylindrical magnet, and the laws of the composition of such 

 streams, we are prepared to investigate the mechanical action of 

 one cylindrical magnet on another. I shall confine myself to the 

 two instances of the disturbance of a moveable magnet by a fixed 

 one, relative to which Gauss has obtained numerical determina- 

 tions. (See Gauss's ' Absolute Measure of the Intensity of Ter- 

 restrial Magnetism/ Gottingen, 1833; and the Annates de Chimie 

 et de Physique, vol. lvii. pp. 56 & 57.) In these experiments the 

 magnets were about a foot long, and the different distances be- 

 tween their middle points varied from four feet to thirteen feet. 

 In both sets the moveable needle when undisturbed was in the 

 plane of the magnetic meridian, the end I have called A being 

 northward, and the end B southward. Also both needles were 

 horizontal with their axes in the same plane. 



In the first set of experiments the axis of the fixed needle was 

 perpendicular to the plane of the magnetic meridian, and pointed 

 to the middle of the moveable needle. Let us take the case of 

 the experiments made when the fixed needle was on the east side 

 of the moveable one, and its end B (from which the current 

 flows) was turned towards the latter. There were three other 

 cases of relative positions of the magnets ; but this one will suf- 

 fice for my purpose. We have next to determine the action of 

 the composite streams on the individual atoms of the moveable 

 needle, so far as such action tends to move the needle as a whole 

 about a vertical axis. The diameter of each needle is supposed 

 to be small compared with its length. 



At the position of any atom of the moveable needle let the 

 velocity of the fluid due to the fixed needle be resolved into u x 

 parallel to the axis of the former, v l perpendicular to this axis, 

 and w x in the vertical direction ; and let u 2 , v 2 , w 2 be the analo- 



