**The** **i** **t** " **power** **of** **that** **appertaining** **to** **a** **x** **and** **b** **x** **multiplied** **by** **the** **j** **t** " **power** **of** **that** **appertaining** **to** **a** **x** **and** **c** **x** **multiplied** **by** &**c**. **If** **any** **two** **of** **the** **linear** **forms**, **say** **p** **x**, **qx**, **be** **supposed** **identical**, **any** **symbolic** **expression** **involving** **the** **factor** (**pq**) **is** **zero**.

**We** **find** **that** **Di** **must** **be** **equal** **to** **p** **x** **g** **x** **for** **then** **t** **x** (**p** **x**) 3 +, **u** (**g** **x**) 3, **Hence**, **if** **px**, **qx** **be** **the** **linear** **factors** **of** **the** **Hessian** 64, **the** **cubic** **can** **be** **put** **into** **the** **form** **A**(**p** **x**) 3 +ï¿½(**g** **x**) 3 **and** **immediately** **solved**.

**Hence**, **from** **the** **identity** **ax** (**pq**) = **px** (**aq**) -**qx** (**ap**), **we** **obtain** (**pet**' = (**aq**) 5px - 5 (**ap**) (**aq**) 4 **pxg** **x** - (**ap**) 5 **gi**, **the** **required** **canonical** **form**.

**On** **the** **other** **hand**, **the** **equations** **q'x** = **q** **and** **yq**' = **q** **have**, **in** **general**, **different** **solutions**.

**Is** **of** **the** **form** **px** 2 + **qx** -{- **r**.

**QX** +**Pu** = **N**.**j** **These** **equations** **are** **applicable** **to** **any** **dynamical** **system** **whatever**.