Note 1 1 : two disks far apart on the same axis 377 



Hence the electromotive force required to produce a spark between convex 

 surfaces, as in Lane's electrometer, is less than if the surfaces had been plane 

 and at the same distance. 



When the air-space is large, the path of the sparks, and therefore the electro- 

 motive force required to produce them, is exceedingly irregular. The accom- 

 panying figure is from a photograph of a succession of sparks taken between 

 the same electrodes from four Leyden jars charged by Holtz's machine. 



A portion of the path near the positive electrode is nearly straight, there 

 is then a sharp turn, which, in all the sparks represented, is in the same direction. 

 Beyond this the course of the spark is very irregular, although its general 

 direction is deflected towards the same side as the first sharp turn. 



NOTE ii, ART. 141. 



Theory of two circular disks on the same axis, their radii being small 

 compared with the distance between them. 



A circular disk may be considered as an ellipsoid, two of whose axes are 

 equal, while the third is zero, and we may apply the method of ellipsoidal 

 co-ordinates to the calculation of the potential*. In the case before us every- 

 thing is symmetrical about the axis, so that we have to consider only the zonal 

 harmonics, and of these only those of even order, unless we wish to distinguish 

 between the surface-density on opposite sides of the same element of the disk, 

 for this depends on the harmonics of odd orders. 



Let a be the radius of the first disk, b that of the second, and c the distance 

 between them. 



We shall use ellipsoidal co-ordinates confocal with the first disk. Let r t 

 and r z be the greatest and least distances respectively of a given point from the 



edge of the disk, and let . 



2 - ~ 2 



...... (2) 



then if z is the distance of the point from the plane of the disk, and r its distance 



from the axis, 



z = a\u>, ...... (3) 



r* = a(i - M 2 ) (p* - I). ...... (4) 



If the surface-density of the electricity on the disk is a function of the 

 distance from the axis, it may be expressed in the form 



a = a + or 2 + &c., ...... (5) 



where a n = --A^P^(^), ...... (6) 



and P 2n is the zonal harmonic of order 2,n. Only even orders are admissible, 

 for since every element of the disk corresponds to two values of fi, numerically 



* See Ferrers' Spherical Harmonics, p. 135. 



