hertz's researches on electrical waves. 161 



that the oscillations are at any given moment everywhere in the same 

 phase. This will certainly be the case in the immediate neighborhood 

 of the primary, and for the present we shall confine onr attention to such 

 ])oints. Let .s- be the distance of a point, measured along the circuit from 

 the air space of the secondary, and F the component e. m. f. at that 

 point along the circular arc d s. Then F is a function of .v, which 

 assumes its original value after passing once round the circle of circum- 

 ference S. It may therefore be expanded iu the form 



2 TT S 2 TT S 



F=A4-B cos g + 4-B'sin— ^+ 



The higher terms of the series may be neglected, as the only result of 

 so doing will be that the approximate theory will give an absolute dis- 

 appearance of sparks where really the disappearance is not quite com- 

 plete, and indeed the experiments are not delicate enough to enable us 

 to compare their results with theory beyond a first approximation. 



The force A acts in the same direction, and is of constant amount at 

 all points of the circle, and therefore it must be independent of the eiec- 

 trostatic e. m. f., as the integral of the latter round the circle is zero. 

 A, then, represents the total e. m. f. of induction, which is measured 

 by the rate of variation of the number of magnetic lines of force which 

 pass through the circle. If the electro-magnetic field containing the 

 circle is assumed to be uniform, A will therefore be proportional to the 

 component of the magnetic induction perpendicular to the plane of the 

 secondary. It will therefore vanish when the direction of the mag- 

 netic induction lies in the plane of the secondary. A will consist of an 

 oscillation, the intensity of which is independent of the position of the 

 air space in the circle, and the corresponding sparking distance will be 

 called a. 



2 7rs 

 The term B' sin ^ can have no eftect in exciting the fundamental 



vibration of the secondary, since it is symmetrical on opposite sides of 

 the air space. 



2 TT S 



The term B cos — ^^^ will give force acting in the same direction iu 



the two quadrants opposed to the air space, and will excite the funda- 

 mental vibration. In the two quadrants adjacent to the air space it 

 will give a force in the opposite direction, but its effect will be less than 

 that of the former one. For the current is zero at the extremities of 

 the circuit, and therefore the electricity can not move so freely as near 

 the center. This corresponds to the fact, that if a string fcisteued at 

 each end has its central portion and ends acted on respectively by 

 oppositely directed forces, its motion will be that due to the force at 

 the central portion, which will excite the fundamental vibration if its 

 oscillations are in unison with the latter. The intensity of the vibra- 

 tion will be proportional to B. Let E be the total e. m. f. in the uni- 

 form field of the secondary, (p the angle between its direction and the 

 plane of the latter, and the angle which its projection on this plane 

 H. Mis. 224 11 



