Approximately Simple Waves. 137 



Again (' Theory o£ Sound/ § 65 a), if 



H = 4cosV, (8) 



we have 



Hcos />£= f cos/rt + cos (p + g)t + cos (p—q)t 



+ icos (p + 2j)* + i cos {p — 2q)t. . . (9) 



If K also be variable and periodic in the same period as 

 H, so that 



K= K 4- K x cos qt + K/ sin </£ 



-r- K s cos 2y* + K 2 ' sin 2<y£ + .. . , . . (10) 



we have the most general periodicity expressed when we 

 substitute these values in (1) ; and the general conclusion as 

 to the periods of the simple vibrations required to represent 

 the effect remains undisturbed. 



If K and H vary together in such a manner that the 

 amplitude y/(K 2 +K 2 ) remains constant, the sole variation is 

 one of phase. My object at present is to call attention to 

 this class of cases, so far as 1 know hitherto neglected, unless 

 an example (Phil. Mag. xxxiv. p. 409, 1892) in which an 

 otherwise constant amplitude is periodically and suddenly 

 reversed be considered an exception. 



If we take 



H = cos (a sin qt), K — sin (a sin qt) , . . (11) 



H and K are of the required periodicity, and the condition 

 of a constant amplitude is satisfied. In fact (1) becomes 



cos (pt — asingf) (12) 



Now, since 



e iacoae =J {J (a) +2/J 1 (a)cos0 + 2i 2 J 2 (a) cos 20+. . . 



+ 2i n J n (a)cGS n6 + . . . , 

 we have 



e ia 9in «* = J + 21 Jx sin qt + 2 J 2 cos 2qt + 21 J 3 sin '6qt 



+ 2J,cos4^+ . . ., .... (13) 

 and thus 



cos (aainqt) = J («) 4- 2J 2 (a) cos 2qt + 2J i (a) cos4g£ + . . ., (14) 

 sin (a sin qt) = 2 J-i(a) sin qt + 2J 3 (a) sin3g£ + . . ., . . (15) 



where J , J 1? &c. denote (as usual) the BesseFs functions of 

 the various orders. In the notation of (3) and (10) 



H 1= H 3 = . . . -0, H/=H 2 '=H 3 '=...=0 3 



H =J H, H 2 =2J 2 («), E i = 2J,{a), &c, 

 k =k 1 =k 2 = . ; . =0, K 2 '=K 4 '= . . . =0, 

 K 1 ' = 2J 1 (a), K,'=2J,(a), K 5 ' = 2J 5 («), &c. 



