164 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
in which the arrow at A lies, and EF that in which the arrow at B lies, the transla- 
tion from A to B consists of two distinct motions ; namely, the shifting of the line 
of direction from CD to EF, and the shifting of the origin along the line of direction 
through a space amounting to GB. The former I shall call lateral and the latter 
longitudinal translation. 
If V denote a directed magnitude which is supposed to be trans- 
lated from A to B, and if u denote the line AB, I shall speak of the 
translation as that of v along u ; a proper mode of expression, be- 
cause every point of the representative arrow v undergoes a motion 
equivalent in magnitude and direction to u. The translation of v 
along u is lateral when the angle at A is 90°, and longitudinal when 0 °. 
( 6 .) Distrihutiveness. — If f (x) be a function of x w^hich possesses the property 
expressed by the equation 
f{x)+f{x')=f{x-\-x'), 
it is said to be a distributive function of x. If x be any number positive, negative, 
integral or fractional, it may be shown from this equation that 
f{x)z=:Qx, 
C denoting a quantity independent of x, namely the value of f{x) when x=l. If x 
be not a number, but some symbol, whether of specific quantity or operation, the 
notation Cx has no meaning recognised in ordinary algebra. Hence, following the 
Well-known precedent of indices*, we may generalize the meaning of Cx by assuming 
it to be the symbolical form for denoting every function, f (a?), which is distributive. 
If we further suppose, that f{x), and therefore C, is a distributive function of 
another independent variable y, we shall find that 
C=C'^ and Cx=.C’xy. 
Thus we may, by the same process of generalization, assume C'xy to be the symbolical 
form for denoting every function of x and y which is distributive with regard to both 
X and y. C' here is manifestly the value of the function when x= l and 3 /=!. Now 
if we adopt C' to be the unit of the function, as, in fact, we do in many cases of 
ordinary products, the symbolic form for denoting the function becomes simply xy. 
(7.) This appears to me to be the simplest and best method of defining the nota- 
tion xy in Symbolical Algebra; though I need not avail myself of it here as it is not 
necessary for my purpose. All I require is some simple notation for denoting a dis- 
tributive function of two variables ; for, as I hope to show, this distributiveness is a 
characteristic of great importance to be distinctly “noted” in the case of the trans- 
lation of a directed magnitude. Now there are three different forms in which a pro- 
duct is written in ordinary algebra, viz. xy x.y and xXy’ of these, the latter two 
are now seldom used, and there is no necessity whatever for this redundancy of 
* has no meaning, according to its original definition, except be a positive integer ; but we give it a 
meaning by defining a to be the notation for every function f (a?) which possesses the property =f{x-\-y)^ 
Fig. 5. 
