of Edinburgh, Session 1870 - 71 . 
501 
3. On some Quaternion Transformations. 
(Abstract.) 
Since the algebraic operator 
when applied to any function of x, simply changes x into x + h, 
it is obvious that if cr be a vector not acted on by 
we have 
-S o-V 
f(p)=f(p+ cr ) > 
e 
whatever function / may be. 
It A bear to the constituents of a~ the same relation as V bears 
to those of p, and if / and F be any two functions which satisfy the 
commutative law in multiplication, this theorem takes the curious 
form 
£ SAV f(p) F (V) = f(p + A) F (V) = F (<r + V)/(p) ; 
of which a particular case is 
/(x)F (2/ ) =/(.« + 4 ) F(y) = F(y + l)f(x). 
The modifications which the general expression undergoes, when 
/ and F are not commutative, are easily seen and need not be 
indicated in this abstract. 
If one of these be an inverse function, such as for instance may 
occur in the solution of a linear differential equation, these 
theorems of course do not give the arbitrary part of the integral, 
The paper contains numerous applications, extensions, and inter¬ 
pretations of these fundamental theorems. 
But there are among them results which appear startling from 
the excessively free use made of the separation of symbols. Of 
VOL. VII. 
