1890-91.] Mr A. M'Aulay on Quaternion Differentiation. 105 
not necessary to say that the term is linear in each of the symbols 
£ and <££, for from the definition of £ it must be so.) If 
<fi<D = — 2/?Swa I 
Q(f,«) = 2Q(a,/J) f ( '• 
If Q be symmetrical in its constituents , and if 
TSm= - |2(/3Swa + aSo>/3) | 
Q(^^) = 2Q(a,/3) / ' • ' ’ ' 
If m have the usual* meaning with reference to the linear vector 
function 
6m = S£ 1 £ 2 £ 3 S<^£ 1 ^>£ 2 ^>£ 3 ..... (9). 
( 10 ) 
From these we deduce that if 
— — S(0 V .(X 
<£ 1 (i) 
3 Y V j V gSwo-jO^ 
SViV 2 V 3 S W 3 >■ . 
( 11 ). 
3 Y O'jO’gScO V ! V 2 
S V 1 V 2 V 3 So- 1 o- 2 o- 3 
If and if/, two linear vector functions of a vector, be connected 
by the equation 
S<££xf = » 
where ^ is a perfectly arbitrary linear vector function of a vector,! 
it follows that 
or, again, if y he self-conjugate but otherwise perfectly arbitrary and 
the above equation hold, it follows that 
<f>— V' 
where 
2cf) — (f) + cf)’ 
and similarly for if/, i.e., </>, if/ are the pure parts of and i/r. 
* Tait’s Quaternions, 3rd ed., §§ 158 et seq. 
t This frequently -recurring and cumbrous phrase is very annoying. Might 
I suggest the term Hamiltonian. Thus, in the present case, we should say — 
“If <f> and i p be two Hamiltonians connected by the equation S 4>(xC=S'l / CxC 
where x is a perfectly arbitrary Hamiltonian,” &c. 
