**We** **have** **A** +**k** 1 **f** =0 2, **O**+**k** 2 **f** = x2, **O**+k3f =4) 2, **and** **Cayley** **shows** **that** **a** **root** **of** **the** **quartic** **can** **be** **xpressed** **in** **the** **determinant** **form** 1, **k**, 0.1y **the** **remaining** **roots** **being** **obtained** **by** **varying** 1, **k**, **x** **the** **signs** **which** **occur** **in** **the** **radicals** 2 **u** **The** **transformation** **to** **the** **normal** **form** **reduces** 1, **k** 3, ?

**One** **of** **these** **contained** **the** **first** **inti** **ation** **of** **the** **achievement** **with** **which** **his** **name** **is** **most** **wid** **ly** **associated**, **for** **it** **was** **in** **a** **paper** **read** **before** **the** **British** **Association** **at** **Cork** **in** 1843, **and** **entitled** "**The** **Calorific** **Effects** **of** **i** **agneto-electricity** **and** **the** **Mechanical** **Value** **of** **Heat**," **that** **he** **xpressed** **the** **conviction** **that** **whenever** **mechanical** **force** **is** **exp** ° **nded** **an** **exact** **equivalent** **of** **heat** **is** **always** **obtained**.