176 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
(48.) It may be well to observe that the following relations result from what has 
been said, viz. 
D(a .^)=7 
D(^. 7 )=a 
D(7.a)=/3 
D(/3.a)=-7 
D( 7 .S) = — a 
D(a. 7 ) = -^ 
whence 
a./3=D *7 
/3.7=D 
7.a=D-'/3 
&c. 
&c. 
&c. 
(49.) Symbol for Units of Longitudinal Translation. — It has been shown that these 
units are all equal to each other ; a unit of longitudinal translation is therefore an 
absolute constant. No distinctive symbol is necessary, therefore, to represent these 
units, and it will be allowable to employ the common unit ( 1 ) for the purpose; just 
in the same way that we denote all units, whether they be linear, superficial, cubical, 
mechanical, by this common symbol. I shall therefore always represent a unit of 
longitudinal translation by 1 ; and thus put 
aXa=l, /3x/3=l, 7 X 7 =!; 
and generally, if m and n denote the magnitudes of u and n, we have (art. 41) 
uXv=nm cos 
6 being the angle made by u and v. 
(50.) Hence if a and 05 ' be any two ‘‘ directions f we have 
aXn'=cosine of angle made by a and a'. 
To this may be added 
magnitude of a.a'=sine of same angle. 
(51.) Projections represented by the lateral and longitudinal translation^products . — 
It is clear from the principles just established, that, if a, /3, 7 denote the directions of 
three coordinate axes, and v any line, the projections of v on the three axes are, nume- 
rically, uxv, I3xv, yXv. 
Again, since mXm is the square of the magnitude of the line u, the projection of v 
on u is, numerically, uxv 
Vuxu 
which, putting for u and v the values xu-^yf^+zy, x'a-\-y'(5-{-z'<y, becomes by longitu- 
dinal multiplication, 
xod yy' zz' 
V x^-\-y'^ + z'^ 
I may use the terms “ lateral and longitudinal multiplication ” to designate the ope- 
rations denoted by M.n and uXv-, for the word “multiplication” has quite lost its 
original and proper signification even in ordinary algebra. 
(52.) If AB represent v, AC the projection of v on u, the mag- 
nitude and direction of u being m and a ; then ^ 
CB=AB — AC=t> — (o5Xt^)a. /\ 
I think CB might be advantageously called the Complement of the / 
Projection of v on u, for CB added to AC makes up or completes v ; 
and thus, employing the usual abbreviation, we may call CB the 
coprojection of v on u. 
