379 
Multiply both members of this equation by A,, and attach 
the symbol =. Then since 
24, Q" = f(Q) =f (w + tx + jy + hz) 
34,(wtry -1)"=f(wt+ry -)) 
DAn(w-ry -1)=f(w-ry - 1) 
we have 
S(w + iv + jy + hz) ct lamin ses Aaa aa 
+ -l1)- - —-1) : 
tials Malia wnt AG tahad Sit tid 5 2 af i yy (ia +jy+khz) (1-) 
the development required. 
PARTICULAR DEDUCTIONS. 
<< There are two particular cases of the above theorem which 
deserve special notice. 
** The first is when #(Q) = Q". ‘The expression given in 
(9) may then be reduced to the following form : 
= (ta + jy + k)) 
(wie + jy kz) = (w? + r°)? (cos nO + 
where 0 = tan? —. 
(2) 
“If n=- 1, this gives e 
(w + tx + jy + hz) = (w? + 7°) (w — ix — jy — kz), 
a well-known theorem. 
The second case is when f(Q) = e®. Here we have 
Ww +7 V1 10 = TV-1 w +7 v-1 Ww -TV-1 
‘pw tiatjytke — e #¢ 2 as 7 : 
é hh ooo Vea Oral (tx + jy + kz) 
td ao 
=e” cosr + ev —— (tt + jy + kz) 
+, ewriatjythe — ew [cosr ae — (1a + jy + kz) | (II.) 
where, as before, 7 = 4/ (a? + y? + 2”). 
