Theory o/'Nobili's Coloured Rings. 11 



querel overlooked this in making his formula, when he re- 

 garded the intensity of the current for each point of the plate 

 as inversely proportional simply to the length of the line of 

 conduction. 



To find the intensity of the current at a given infinitely 

 small point of the conductor, we have first to calculate the 

 resistance which the space between the curves of the current 

 circularly bounding this spot offers to the current. In the 

 present instance, however, the resistance of the curvilinearly 

 bounded portion of this space in comparison with that of the 

 part rectilinearly bounded may probably be neglected, be- 

 cause, as we have mentioned, in the former, which of itself 

 is much shorter, the curves of the current also separate from 

 each other, so that the section of the path of the current is 

 here much greater. Thus, according to M. BecquerePs no 

 longer justified process, the conducting rays Os f Os' may be re- 

 placed by Ors, Ors 1 , the true curves of the current; or, neglect- 

 ing the resistance of the portion rs, rV, the triangle Oss 1 by the 

 triangle Orr', with tolerable accuracy, but then, although 

 starting from the same point of view, we must alter the ex- 

 pression of the resistance, whicli a partial current has to over- 

 come. 



We must then consider here as the partial path of the cur- 

 rent, the space between two conical envelopes, the axis and 

 apex of which coincide with each other, as also with the centre 

 O of the point from which the current escapes, and its perpen- 

 dicular distance OO' from the plate, which latter forms the 

 common basal surface of the cone, and the very obtuse angles 

 of which are separated at the apex by an infinitely small angle 

 2<p. The resistance of such a space is easily determined. If 

 we denote the distance of any point of the conical envelope from 

 O by£, the angle at the surface of the base byy, and the inverse 

 value of the resistance for the unit, the length and the section 

 of the path by w, the element of this becomes 



. 1 d| 



dw = - j .-T5-. 



2ttk> cos ytg<p ^ 



The integration may be completed between the limits £=S = 

 the total length of the line of conduction, as the upper, and 

 Z=p, any small, but finally constant quantity, as the lower 

 boundary. Thus by p we must understand the diameter of 

 the hemispherical point from which the current escapes, for 

 if we neglected it, as in M. Becquerel's process, the resistance 

 would become infinite. We have 



Was 



J»-P 



27roop "a cos ytg<p' 



