There are three successive substitutions for each of these equations. The formulæ (2.), (3.), and (4.) are implicitly contained in (1.), which latter we may consider as being in fact the condensed expression of any of the former. It will be observed that every succeeding substitution must contain twice as many
’s as its predecessor. So that if a problem require
substitutions, the successive series of numbers for the
’s in the whole of them will be 2, 4, 8, 16 ...
.