180 



PROFESSOR K. PEARSON AND MISS A. LEE ON THE VIBRATIONS 



unity seems finally to have mastered his experimental suspicion as to the possible 

 coexistence of two waves in this plane (see our p. 177). Accordingly we have worked 

 out both the velocity and phase of this compound wave, which at the same time 

 determines the total transverse wave (see p. 170), in order that a comparison may be 

 made with HERTZ'S results for an undamped single wave in the equatorial plane. 

 Returning to the formula (x) and putting 6 = %ir, it may be read as : 



where if = 



Hence, 



From this we deduce : 



1 



2rr/\ ' 



Z (r) sin ft = (2 tan x - f ) El (2ir/X) 

 Z (r) cos ft = ( 1 tan 2 x + tan x f) 



(xxxviii), 



(27T/X) 3 



(xxxix). 



tan(ft-2 x ) = 



cos 2 



lirr 



sin 2y. 



zirr 



~ i si" 2* ) - cos* x (cos 5 x - I in ! Y.) 



where 



c = cos 2 x, o = sin 2x, c 2 = cos 2 x (cos 2 x I sin 2 x). 



This formula is fairly well adapted for logarithmic calculation. The phase may be 

 obtained by taking 



84 = { - A- 



It may be compared with the value for 8,, obtained from ,, on p. 172, where we 

 easily find 



tan (ft -2x) =-_ 



- 3cos l y(- - - sin2y 



' \ - 



- | sin 2 x j - 3 cos 2 x (cos 8 x - } sin 2 x ) 



if 



(xli), 



As before 



d = 3 cos' 4 x (cos 2 x - i sin 2 x ). 



Si = - ft. 



This again readily lends itself to logarithmic calculation. 

 Both have for asymptote the same 45 line, namely, 



S = { - Or + 2 X ). 



