186 Prof. Challis on the Hydro dynamical Theory of the 



occupation of space by the atoms towards the denser end. Now 

 we may conceive this mean effect to result from the separate 

 effects of a vast number of atoms contained within a thin trans- 

 verse slice of the cylinder, inasmuch as the individual motions 

 due to the occupation of space by the atoms may coexist, and 

 the parts of the motions resolved transversely to the axis will in 

 that case destroy each other. Also it is to be considered that 

 the motions of the aether resulting from the mean of the impulses 

 must satify the condition of circulating. 



14. This being understood, it will be seen to be allowable 

 to substitute for the impulsive effect of contraction of channel 

 that of a motion forward in the same direction of the aggre- 

 gation of atoms contained in the above-mentioned slice, the 

 fluid being relatively at rest. For on this supposition there 

 will be a mean impulse parallel to the axis of the cylinder, 

 which will be the sum of the impulses of the individual atoms 

 resolved in that direction, and moreover will give rise to a 

 circulating motion. The last assertion rests on Poisson's so- 

 lution of the problem of the simultaneous motions of a ball-pen- 

 dulum and the surrounding fluid, according to which the lines 

 of motion of the fluid are reentering ; and this being the case 

 with respect to each atom, the result of the composition of all 

 the motions will be circulating motion. Now, assuming the 

 transverse section of the c}dinder to be small, it is evident 

 that the stream resulting from the action of all the atoms in 

 the slice will have quam proxime the same form as that pro- 

 duced by a single atom situated at the middle point of the slice. 

 But by Poisson's solution we obtain the analytical expression 

 of the motion of the fluid in this case. Hence a formula for 

 expressing the motion due to all the atoms in a given small 

 slice may be at once inferred. 



15. Let A and B be the extreme points of the axis of the 

 cylinder, its middle point, P any extraneous point the coor- 

 dinates of which reckoned from O along and perpendicular to 

 the axis are p and q, and Q being a point of the axis distant by 

 a: from ; let the straight line joining P and Q make an angle 6 

 with the positive direction of the axis. Then if PQ = r, fi be 

 the velocity of the atom, and a its radius, by the above-men- 

 tioned solution the velocity at P in the direction from Q to P is 



~- cos 6, and that perpendicular to P Q tending in the negative 



UiO? 



direction is ^~ sin 6. Hence, denoting by X and Y the total 

 &r 



velocities resolved along and transversely to the axis, we have 

 X= &£ cos 2 6>- ^sin 2 0, Y= ^cos 6 sin 6> + ^sinflcostf; 



