Heterogeneous Function
Summary: A function of is heterogeneous if its terms have at least two different degrees. Euler classifies such functions by the number of distinct degrees: bifid (two), trifid (three), and so on. Some heterogeneous functions can be made homogeneous by a substitution of the form or ; no general criterion is given.
Sources: chapter5
Last updated: 2026-04-23
Classification by number of distinct degrees (§92)
A bifid function has terms of exactly two different degrees — it is a sum of two homogeneous pieces. Example (source: chapter5, §92):
is bifid: the parts and have degrees 5 and 2 respectively.
A trifid function has three distinct degrees — a sum of three homogeneous pieces. Example:
has degrees 6, 4, 1.
Not every rational or irrational function splits cleanly into homogeneous parts. Euler’s examples of the “cannot be resolved” kind:
For these the degree is genuinely not defined.
§93 — Reducing to homogeneous by substitution
Sometimes a substitution for one variable converts a heterogeneous expression into a homogeneous one. Euler gives two examples with no general theory (source: chapter5, §93):
Example 1. . The substitution gives
which is homogeneous of degree 5 in .
Example 2. . The substitution gives
homogeneous of degree 1.
Euler’s comment: “there are much more difficult cases which can be reduced to homogeneity, but with substitutions which are not so simple.” No criterion, no algorithm.