352 Prof. Challis on the Hydrodynamical Theory of the 



It is here supposed that the direction of the motion about the 

 axis is such that at the centre C, where the two motions have 

 the same direction the compound motion is in the positive di- 

 rection; for since at that point p — z and q=0 } the formulae 



give X = 0, and Z= + ^- 



46. The above values of X and Z determine the direction the 

 needle would take supposing it to be acted upon solely by the 

 two motions considered in the preceding argument. To find the 

 effect of a coil composed of any number of such circular rheo- 

 phores all of the same magnitude, and having the axis of z for 

 their common axis, it is required to find ^Xdz and {Zdz, the in- 

 tegrals being taken from one end to the other of the axis of the 

 coil. If the length of the axis be 2/ and its middle point be at 

 the origin of coordinates, the limits of the integration are 

 2= — I and z= -\-l. Hence substituting, for the sake of brevity, 



n } for ((j,-J)i+ {q-aY)-\ n 3 for ([p-lf+ (q + a)*y*, 



rc 2 for ((^ + /) 2 +fe-fl) 2 ) _i , n 4 for (> + /)*+ {q + a)*)~K 



it will be found by effecting the integrations that 



k§Xdz=kfi(n J — n^— n 3 +n 4 )j 



Jc\Zdz=kfM[- n.— w 2 — — ~-n 3 + ±— — n 4 ), 



J ^\q — a l q — a * q + a d q + a V 



k being the constant factor which converts the sum of the velo- 

 cities into the sum of the directive forces. 



47. If instead of a single cylindrical coil we have any num- 

 ber of such coils in juxtaposition and having a common axis, 

 the directive action of this compound coil would be found by ob- 

 taining the integrals 



§(k$Xdz)da, §{k§Zdz)da, 



between the limits of the least and greatest values of the radius 

 a. Supposing a l to be the mean value of a, and the least 

 and greatest values to be #, — e and a x + e, the results of these 

 integrations, if e be very small compared with a v would be very 

 approximately obtained by multiplying the previous integrals by 

 2e, putting «, in the place of a, and altering the constant k. 



48. The direct action of the transverse circular motions on 

 the magnet having thus been calculated, we have now to take 

 account of an indirect action, the hydrodynamical origin of 

 which I proceed to explain. From hydrodynamics it is known 

 that the steady motions of a mass of fluid of unlimited di- 

 mensions, whether it be incompressible or highly elastic, may 

 be supposed to coexist, and that, if U, V, W be the sums of 



