of Edinburgh, Session 1881-82. 
351 
We have also, when 
(<£-</>> = Vep, 
e = V (aoq + /?/5 1 ) . 
Squaring e and subtracting it from R 2 , we get, when all is de- 
veloped : 
R 2 — e 2 = T 2 . acq 4 - T 2 . j3/3 1 
+ 2Saa 1 S j3/3 1 — 2Sa/$ 1 Scq/3 
-1- 2Sa/5Sa 1 /S 1 
+ dSa/^Scq/I — 4:^>aaf>/3j3 1 . 
The five double products of scalars reduce themselves to 
2Sa[/8Scq/3 1 — cqS /3/3 1 + /^Scq/?] 
= 2SaV/3a 1 /3 l 
= 2T(aa 1 )T.( / g/3 1 )S.UaU/5Ua 1 U/? 1 . 
We may put 
S.UaU/TlJajU/^ = - COS 0), 
because the absolute value of this scalar cannot outpass unity. 
Thus, putting 
we have 
R;j = T 2 . acq — 2T . acqT . j3/3 1 COS w + T 2 . > 
R 2 = R§ - T 2 . e . 
This expression of R 2 shows that when Te = 0, namely, when <f> is 
self conjugate , then the value of R 2 is essentiallg positive, — Ejq and 
the roots h are real h priori. It is true that when Te is comprised 
within the proper limit, then R 2 remains positive, so that the roots 
h and the solutions o- may remain real, even when </> is not self- 
conjugate, provided : 
Te < TR 0 . 
§ 5 . We will apply our equations ( 14 ) and ( 15 ) to two correspond- 
ing cases; the first to the case of the example of § 159 of the 
Elementary Treatise on Quaternions ; the second to the case of the 
construction of equation (8) of Professor Tait’s paper on Minding'’ s 
Theorem {Trans. R.S.E. 1880 ). 
The Example of § 159 demands the solution of 
V.ep = S. 
(j>p = Yep . 
We put 
