440 Prof. Challis on the Hydrodynamical Theory of the 



It is not necessary for my present purpose to go through the 

 details of integrating, which would be a process of considerable 

 complexity ; the following deduction from the differential quan- 

 tity itself will suffice for indicating, for any given inclination of 

 the axes of the rheophores, the character of the action of the 

 fixed rheophore which tends to cause rotation of the movable 

 one. 



71. It may, first, be observed that that differential quantity 

 has the same sign for all values of the angle of inclination of the 

 rheophores from co = to o) = 7T, because 6 has the same sign in 

 all these positions. According to the assumed directions of the 

 currents relative to the point 0, V and V and, by consequence, 

 i and i' have different signs ; and the sign of the differential 

 is therefore negative. This sign is to be taken as indicating re- 

 pulsion, because, in fact, for all the values of co the parts of two 

 elementary currents resolved transversely to the line joining 

 them will be in opposite directions. This would also be the case 

 if the directions of V and Vbe both changed, so as to be still 

 opposite as regards the point 0. But if the currents be both 

 from, or both towards, that point, so that V and V have the 

 same sign, the sign of the differential quantity would be posi- 

 tive, and the action between any two elements would be attrac- 

 tive. 



72. Taking a case in which co is very small, if the distances 

 s and s 1 of two elements from be nearly equal, r 2 would be very 

 small, being nearly equal to (s — s 1 ) 2 , and at the same time 6 and 

 6 1 would not differ much from right angles. Hence, by the 

 formula, the force between two elements so situated, whether 

 attractive or repulsive, would be very great, and, if the value of r 

 could actually be zero, would become infinite. Again, supposing 

 co to be nearly equal to it, r would be nearly equal to s + s 1 , 

 6 and 6' would each be very small, and consequently the action, 

 whether attractive or repulsive, would be very small between 

 any two elements. This action would change sign with the 

 change of sign of 6, and, therefore, if repulsive, it would be 

 directive. Prom these considerations it may be concluded that 

 in the case of repulsive action (for which V and V have different 

 signs), the fixed rheophore tends to make the other revolve from 

 the position of maximum repulsion and unstable equilibrium, 

 for which co — Q, to the opposite position of minimum repulsion 

 and stable equilibrium ; and in the case of attractive action 

 (for which V and V have the same signs), the fixed rheophore 

 tends to produce a revolution of the other from the position of 

 minimum attraction and unstable equilibrium, for which co = 7r, 

 to the opposite position of maximum attraction and stable 

 equilibrium. These results agree with well-known facts. 



