THE CALCULATION OF MODULATION PRODUCTS 243 



similar rule for the subscripts n, ii — \, n — 2, valid when (30) is used. 

 For example, by direct application of (30), we deduce that 



_ _ 2[(1 - m)k^ + w + l]^i-„t, n+i-\- (n — m— l)kA-m. n 



Now since it is known that we may replace the subscripts 2 — m, 

 \ — m, and —m by w — 2, m — 1, and m respectively, we may show 

 that 



_ _ 2[(W — 1)F — n — \^Am-l,n+i -\r {m — n -\- S)kAm-2,n+2 



'^'"" ~ (m - n^ \)k 



which is exactly equivalent to the relation we get if we replace nhy —n 

 throughout in (30) and then substitute A^.-n = A„in, Am.-n-i = 



Am, n+1) Am, -n-2 — Am, n+2- 



It may be remarked that it would be incorrect to base a proof of 

 interchangeability of sign of subscripts on (27) because the equivalence 

 of (27) and (28) has not been demonstrated for a sufficient range of 

 values of m and n. 



