438- Prof. Cliallis on the Hydrodynamical Theory of the 



this problem may be given as follows on the hydrodynamical 

 theory. ■ 



69. From what is said in art. 67, it is not allowable in that 

 theory to make use of any formula for the value of F differing 

 from the formula (c) obtained in art. 63. For shortness, i and i 

 will be respectively put for yu,V and fJV. Then, since for the 

 action of the fixed rheophore on the movable vertical sides of 

 the other we have for any two elements 6' = tt — 6 and e = Q, the 

 formula becomes 



Y=^llsmWdsds. 



To obtain the resultant of the action of the indefinite current 

 on one of the movable sides of the other, it is only required to 

 calculate S . F sin 6, the sum of the transverse actions, because, 

 evidently, the resultant in the direction parallel to the indefinitely 

 long current will be zero. The action in this instance is repul- 

 sive, because, by hypothesis, the currents are in opposite direc- 

 tions. The summation X . F sin 6 is required to embrace the 

 transverse repulsive action of every element of the fixed rheo- 

 phore on every element of the movable one. Consequently 



2 . F sin fl=ff E ° ? / sin 3 6 ds ds 1 , 



"F# 



the integrals being taken between assignable limits. Let the 

 rheophore acted upon be that whose length is / and distance 

 from the fixed rheophore a, and let ds' be one of its elements. 

 Then, supposing s to be reckoned from the foot of a perpendi- 

 dicular to the fixed rheophore drawn from this element, we 

 shall have s= a cot 6; and since the distance (r) between the 

 two elements is a cosec 6 3 we get by substitution in the for- 

 mula, 



2 . F sin = f f- Ml sin 3 dd ds 1 , 



the first integral being taken from = to = 7r, and the other 

 from s' = to $' = /. The result will therefore be 



X.F. in 0= -!&«£/.' 



c oa 



By a like process we should obtain for the repulsion on the other 

 vertical portion of the movable rheophore 



• . ■ - oa! . 



And since in case of equilibrium these forces are equal, it follows 

 that I __ V I _ a 



