352 KENNELLY-NABESHIMA— ESTABLISHING A 



size equal to that of OB, as obtained by consulting the stationary- 

 vector diagram Fig. 15. OB is now the new rotating vector, but 

 its projection on OX must commence at the same value Ob as 

 existed at the instant of the second reflection's arrival. This re- 

 quires an instantaneous leap of the projecting vector from OB' to 

 OB. The new vector OB now executes the half circle BC (187°) 

 at the standard angular velocity, until it reaches the position OC, 

 8.6 miUiseconds later than that it occupied at OB. Here the third 

 reflection arrives and requires an instantaneous change of the vector 

 from OC to OC, with a common instantaneous projection Oc. 

 The new resultant vector OC, of three reflections, now rotates 

 through 187° to OD', when the fourth reflection arrives and causes 

 an immediate change to OD. This process goes on indefinitely. It 

 will be observed that although Fig. 17 represents the instantaneous 

 e.m.f. at the receiving end of the line from the moment of arrival 

 of the first reflection, the angular position of the rotating vector 

 ceases to correspond to the time which has elapsed, owing to the 

 instantaneous backward leaps at the moments when new reflections 

 arrive. 



It may be observed that, referring to Fig. 15, the envelope of the 

 successive resultants of e.m.f. reflections Oa, b, c, etc., is an equi- 

 angular spiral. The same is true for the corresponding envelope of 

 current-wave reflections, as laid down in a stationary vector diagram 

 A B C, etc., Fig. 15. When the successive waves of arrival and 

 reflection are summed at a point P along the line, instead of at the 

 distant end B, the same propositions will be found to apply to the 

 envelope of the successive reflections. The vectors of arriving 

 waves will, however, differ, in general, from those of the reflections 

 succeeding them. 



The algebraic expression corresponding to Fig. 17 is: 



€£ = EAm{me-^' sin (cot - 62) + m(i - m)r^^' sin (co/ - 3^2) 



-h w(i - w) V^^' sin (00/ - 5^2) + • • • } inst. volts. (i8a) 



, Here each term must be included after the time has elapsed for 

 its arrival. Three phase angles present themselves in each term; 



