1901—2.] Late Professor Tait on Quaternion Notes. 345 
The remaining notes should be read in connection with former 
brief abstracts read before the Society (May 18 and June 1, 1896 ; 
March 1, 1897 ; May 1, 1899). In the first line D is a contraction 
for Sa fiy, so that in the expanded form we have 
<£pSa/?y = ^aS/fyp + g. 2 fiSyap + g s ySa/3cf> 
where a/3y are the directions of the real roots g 1 g 2 g 3 . 
This may be written in the form 
D.</>p = (AaSa. + B/3S/3. + CySy.)(A' V fiySfiy. 4* . . . )p 
= D(AA'aS /3yp+ . . . ) 
= u)Top 
where 
(op = AaSap + (3 p + . . . 
and 
TSp = A'Vfiy&fiyp + BWyaSyap + . . . 
both being two self-conjugate functions or pure strains. 
Expanding and comparing term by term, we find 
D.AA' = p 1} D.BB' = 0 2 , D.CC' = p 3 . 
When ABC are given, A'B'C' follow. Hence there is an infinite 
number of ways in which (with real roots) may be decomposed 
into two pure strains. 
The next section begins with 
cf)p = oi& p (two pures) 
~9P 
then 
so that the roots are being sought. 
Put 
6T^p = (T 
to£7“cr = gZ5~ }i (r 
Z!3^u)ZlJ\ a = gcr 
Hence G7*oou7- has the same root values as wS7, but different 
directions. 
The linear vector function is now expressed in the form 
</>p = C7p -f Vep 
where 67 is the pure non-rotational part of (f > . 
PROC. ROY. SOC. EDIN. — YOL. XXIV. 
23 
