238 Dr. J. A. Fleming on the Distribution of 



formula agrees, as it should do in these reduced cases, with the 

 results of the direct method based on first principles. If a 

 value of be found which will make the expression 



27T + 27T0-20 2 



equal to it + 4, then for such a position of the diameter AB 

 relatively to PQ the resistance of the circle and its diagonal 

 PQ would be exactly equal to the resistance of half the dia- 

 metral wire or to its radius, assuming both the circle and 

 diagonal to be made of wire of equal conductivity per unit of 

 length. To find the value of for which this is the case, we 

 have to solve the quadratic 



27r + 27r0-2<9 2 :=7r-i-4. 



If we put — tqT\ x °} where x° is the number of degrees equi- 



valent to the angle 0, we find, as a solution for this quadratic, 

 that the positive root is nearly 



171°°804. 



Now 3 radians, or 3 unit-angles in circular measure, are nearly 



171°-887. 



Hence, for a position of the diagonal PQ as in fig. 14, when 

 the arc AP is nearly equal to 7r — 3, or to the fractional part 

 of 7r, the resistance of the circle and diagonal PQ measured 

 between the points A, B is very nearly equal to that of half the 

 diagonal PQ; or, which is the same thing, the resistance of PQ 

 alone is nearly double the combined resistance of the circle and 

 diagonal measured between the points A and B at the extremity 

 of a diameter removed 171 o, 804 from PQ. 



§ 12. A small practical application of this last example may 

 be made in constructing a variable resistance. 



Let PAQB (fig. 15) be a narrow circular canal cut in a slab 

 of wood or ebonite and filled with mercury. Let PDQ be a 

 bent copper wire balanced on a pivot CD, and having its ends 

 P and Q dipping in the trough at opposite extremities of a 

 diameter of the circular trough PAQB. 



The total resistance between any two points A and B in the 

 trough, which are also diametrically opposite, can be varied 

 within limits by changing the position of PQ relatively to AB. 

 When PQ is turned so that it is at right angles to the dia- 

 meter AB, it does not affect the total resistance between A 

 and B, and maybe removed. The resistance is then just that 

 of the circular band of mercury taken at opposite extremities 

 of its diameter. When PQ is coincident with AB it reduces 

 the resistance, and in intermediate positions the joint resistance 



