170 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
(22.) It is important to observe, here, that the + in u-^-n! denotes successive, 
while that in v-\-v' denotes simultaneous addition. 
(23.) Notation adopted to represent the two effects. I have stated above the 
reasons why u.v and uxv may be appropriated to denote, simply, any distributive 
functions of u and v. I have here shown the existence of two such functions, very 
important to be “noted” symbolically, and to be distinguished from each other. I 
shall therefore venture farther to employ the notation u.v exclusively for the pur- 
pose of representing the lateral effect of the translation of v along w, and the notation 
uXv exclusively to represent the longitudinal effect. 
(24.) As regards the order of the factors, I shall always suppose that the second 
factor is the translated magnitude, and the first factor the line along which it is 
translated. 
(25.) In using these notations I am not warranted to attribute to them, without 
proof, any property of an ordinary product, except its distributiveness : for example, 
I must not put u.v~v .u, without investigating whether this equation holds as regards 
the effects represented by u.v and v.u. Nor again, if m and n be any numbers, can 
I, without proof, put {mu) .{nv) = mn{u.v). These points I shall now consider. 
(26.) May Numerical Coefficients, occurring in u.v or uxv, he brought out and 
incorporated by actual multiplication } — Supposing m and n to denote pure numbers, 
may we put {mu) .{77v)=mn{u.v), and {mu)x{nv) = mn{uxv)'i Or, to express the 
question in words, is the effect produced by the translation of nv along mu equivalent 
to mn times the effect of the translation of v along m? It is very important to bear 
in mind, as regards this question, that nv means v-\-v-\-v-{- &c. ^f ut together" by 
simultaneous addition; while mu means u-\-u-\-u-\- &c. put together" by successive 
addition (see art. 22). Hence it will not be difficult to show that u.{nv)=n{u.v) 
in virtue of the distributive property; but, that some additional consideration is 
requisite to determine whether {mu) .v=m{u.v). 
(2/.) First, as regards u.{nv). The nv^ here have the same origin 
A, and they are translated simultaneously from A to B. This trans- 
lation is manifestly the same as if each v were translated separately 
from A to B : and thus it follows that the translation of nv along u 
is the same thing as n translations of v along u ; or, in symbols, 
u.{nv)=n{u.v), and uX{nv)=n{uXv). 
Indeed this is nothing more than a re-assertion of the distributive nature of the 
translation of v along u, as regards v. 
(28.) Secondly. In the expression {mu).v the have not the same origin, but are 
put together'" successively, as is represented in fig. 16, pjo- 16 . 
making up the line AB (supposing, for a moment, that 
m—b). It is clear, then, that {bu).v means a transla- a.^ — ^ n 
ZO W U' u- 
tion of V from A to B, while 5 {u.v) means five transla- 
tions of V from A to A'. Hence, before we can decide whether {mu) .v=m{u.v), we 
