PHYSICAL MEASUREMENTS OF AUDITION Ml 



where the coefficients ao, a u a» . . . belong to the expansion of the 

 function into a power series. Now if 8p is a sinusoidal variation then 



8p=p cos cct (3) 



where — is the frequency of vibration. Substituting this value in 



(2), terms containing the cosine raised to integral powers are ob- 

 tained. These can be expanded into multiple angle functions. For 

 example, for the first four powers 



cos 2 oj/ = ^ cos 2 00/ -f-^, (4) 



cos 3 cot — j cos 3 co/ + f cos cat, (5) 



cos 4 oj/ = | cos4co/ + ^ cos 2 w/ + i- (6) 



It is evident then that the displacement X will be represented by a 

 formula 



X = 6o + &lCOSOj/ + ^2COS 2co/ + 6 3 cos3 co/+ .... 



In other words when a periodic force of only one frequency is im- 

 pressed upon the ear-drum this same frequency and in addition all 

 its harmonic frequencies are impressed upon the fluid of the inner ear. 

 If two pure tones are impressed upon the ear then hp is given by 



bp = pi COS (Cxt-\rp2 COS b)2t. 



If this value is substituted in equation (2), terms of the form cos n caj 

 and cos m u 2 t and cos" (a x t cos m w 2 / are obtained. The first two forms 

 give rise to all the harmonics and the third form gives rise to the 

 summation and difference tones. For example, the first four terms are 



a = a 



a\hp = a\{pi coso)i/+^ 2 cosco 2 



ai{bpy = a 2 [\ p\ cos 2 wi/ + 1 p\ cos 2 <aot-\-pip 2 



JCOS ( Wl -C0 2 )/ + COS («x+«*)<{ -H(tf+$)] 



a 3 (8p) 3 = a 3 [ ( j p\+ -| pi p\) cos Wl *+ j p\ cos 3 u x t + 



( 3 3 \ 1 3 



( -j P\+ 2" Pi Pi) cos w 2^+ ~r Pi cos 3 w 2 /+ — $ p 2 cos (wj/4-2 «i*) 



3 3 



-\- — pi pi cos (<a%t — 2 coi/) + — £i^cos (aji/ + 2co 2 /) 



3 ~i 



-f — £i$cos(«i/-2w 2 • 



