379 
Multiply both members of this equation by 4,, and attach 
the symbol =. Then since 
24, Q" = f(Q) =f (w t+ ta + jy + kz) 
=A, (wiry -1yf"=f(wt+ry -1) 
ZAn(w-ry -1)=f(w-ry - 1) 
we have 
S(w + iw + jy + kez) MACE RITE RCA 
pt raf 2A) Cod wy DRE 
aoe, v LO aTV AD) Cie + jy the) OL) 
the development required. 
PARTICULAR DEDUCTIONS. 
‘‘ There are two particular cases of the above theorem which 
deserve special notice. 
“‘ The first is when f(Q) = Q". The expression given in 
(9) may then be reduced to the following form : 
Smt (ie + jy + &:)) 
(w + ix + jy + hz)" = (w? +r)? (cos nO + 
where @ = tan —. 
Ww 
“If n=— 1, this gives i 
(w + i + jy + kz) = (w* + 2°) (w — ix — jy — kz), 
a well-known theorem. 
The second case is when f(Q) = e®. Here we have 
Ate EW srv-l 4 ew-TH-1 ewtTvl _ ew-ri-1 
LE PY 7 kc ae A Bite SF he 2 teen et EC 9 
ew ria jys ke — 9 + yal (12 + jy + kz) 
TG oe ae 
ed cost He (1@ + jy + kz) 
MO TN 
*. ewtwrigythe — e [eosr + a (ix + jy + kz)| (IL) 
where, as before, 7 = + (a? + y* + 2). 
