484 Prof. Challis on the Hydrodynamical Theory of the 



In Jamin (p. 201) the sinuous and return currents are repre- 

 sented by a figure as being connected, but movable about an axis, 

 and the other rheophore as fixed. In this case let the velocity W 

 within the sinuous rheophore and parallel to its axis, be resolved 

 at any point into W cos a parallel to the current's mean direction, 

 and W sin «. in the plane through the point transverse to that 

 direction, and let w be the velocity at the same point due to 

 the current of the fixed rheophore. Then the velocity W sin a, 

 as being transverse to the axis of motion, will have no effect ; but 

 the composition of W cos a and w will give rise to atranslatory 

 action on the element of the sinuous rheophore which, by the 

 same reasoning as that in arts. 55 and 56, may be expressed 

 by 



— W cos « — - ds, 

 dr 



ds being the length of the element, and r the distance of the 

 given point from the axis of the fixed rheophore. Now, as W is 

 the same for the sinuous current as for the return straight one, 

 and ds cos a is the projection of the element ds on the latter, it 

 follows that the sum of all the elementary actions is the same 

 for both rheophores, but in opposite directions, and that Con- 

 sequently, as is found by experiment, no motion takes place. 



61. We may now proceed to obtain a general formula for the 

 action of an element of one rheophore upon an element of 

 another, as due to the hydrodynamical action of the corre- 

 sponding elements of the currents pertaining to the two rheo- 

 phores. Let ds be the length of the element of one of the 

 rheophores, and ds J the length of the element of the other ; and, 

 supposing their axes to have any relative positions, let their 

 middle points be joined by a straight line of length r. Also 

 let 6 and 1 be respectively the inclinations of the axes to the 

 joining line, and conceive the plane containing the element ds* 

 and that line to be inclined by the angle e to the plane contain- 

 ing ds and the same line. Hence, V and V being the velocities 

 of the currents at the middle points of the elements, the re- 

 solved parts along the joining line are V cos 6 and Y' cos &, and 

 those perpendicular to the same are V sin 6 and V sin 6 l . The 

 resolved part of the last of these velocities in the plane containing 

 the velocity V sin 6 is Y' sin & cos e; so that the velocities V sin 6 

 and V sin 6' cos € are parallel to each other and perpendicular to 

 the line of junction. The resolved part perpendicular to the same 

 plane is V sin e. 



62. It is evident that the velocities V and V may be taken to 

 represent the intrinsic intensities of the two currents. Hence 

 Y ds and Y 1 ds' will be respectively proportional to the products of 



