1 **exp**(**Adlo** + **vdol**) = (1+/oD10+ **v** **Doi** +..ï¿½+ **VQ** +.ï¿½.)**f**; **now**, **since** **the** **introduction** **of** **the** **new** **quantities** 1.1., **v** **results** **in** **the** **addition** **to** **the** **function** (plglp2g2p3g3...) **of** **the** **new** **terms** **A** **PI** Pg1 (**p** 2q2 **p** 3g3ï¿½ï¿½ï¿½) +/ AP2Pg2 (**p** 1 **g** 1P343 ...)+/ Z3vg3 (**p** **l** **g** **i** **p** 2 **g** 2 ...)+ ï¿½, **we** **find** DP141(plqip2q2p3q3ï¿½ï¿½ï¿½) = (**p** 2 **q** 2 **p** 3 **q** 3ï¿½ï¿½ï¿½), **and** **thence** **D** P141 **D** P242 **D** P343 ï¿½ï¿½.

**Putting** **q**=**a**+,61+**yj**+**bk**, **Hamilton** **calls** **a** **the** **scalar** **part** **of** **q**, **and** **denotes** **it** **by** **Sq**; **he** **also** **writes** **Vq** **for** 01+**yj**+**b** ï¿½, **which** **is** **called** **the** **vector** **part** **of** **q**.

**Thus** **every** **quaternion** **may** **be** **written** **in** **the** **form** **q** = **Sq**+**Vq**, **where** **either** **Sq** **or** **Vq** **may** **separately** **vanish**; **so** **that** **ordinary** **algebraic** **quantities** (**or** **scalars**, **as** **we** **shall** **call** **them**) **and** **pure** **vectors** **may** **each** **be** **regarded** **as** **special** **cases** **of** **quaternions**.

**If** **we** **put** **qo**= **Sq**' - **Vq**', **then** **qo** **is** **called** **the** **conjugate** **of** **q**', **and** **the** **scalar** **q'qo** = **qoq**' **is** **called** **the** **norm** **of** **q**' **and** **written** **Nq**'.

**Vq**/**Sq**, **and** **that** **of** **the** **foot** **of** **perpendicular** **from** **centre** **on** **plane** **is** **Srg** **i**.

**Sq**/**Vq**, **the** **product** **being** **the** (**radius**)2, **that** **is** (**Srq** 1) 2.