OF THE TRANSLATION OF A DIRECTED MAGNITUDE. l75 
that is, the directrix of the sum of the two translations is the sum of their direc- 
trices. 
(43.) Generally, it is manifest from this result, that the Directrix of the Sum of 
any number of Translations is the Sum of their Directrices. 
It is not necessary to point out the importance of this rule as regards the summa- 
tion of translations, nor its identity with the well-known rule in the Theory of 
Couples. 
(44.) For the sake of convenience it will be worth while to employ some abbre- 
viated mode of specifying the directrix of any translation ; for this purpose I shall 
adopt the following notation, which, it will be found, will answer all purposes, and 
at the same time very distinctly mark that property in which the peculiar relation be- 
tween a translation and its directrix consists. The property I allude to is the theorem 
just proved in article 42. 
I shall employ the letter D to stand as an abbreviation for the words ‘‘directrix off 
and thus D{u.v) will mean the directrix of the translation u.v. It will be borne in 
mind, then, that D{u.v) denotes a line at right angles to the plane of u.v, and con- 
taining as many units of length as there are units of area in the parallelogram con- 
structed on u and v. 
(45.) Distributiveness of the Operation thus represented. — It is most important to 
notice the distributive nature of the symbol D. By art. 42, we have immediately 
D{u.v)-\-D{u' .v') = D{u.v-\-u’ .d) ; 
whence, D denotes a distributive function. 
(46.) Consequences hence resulting. — From the equation in art. 45 it follows, that^ 
if denote any numerical coefficient, positive or negative, 
D{mu .v) — mD{u .v) . 
Again, since 
D{u! .v') — D{u.v)=D{u! .v' — u.v), 
we have, passing to limits, 
d{D{u .v)^=D{d{u .v'f ) ; 
whence also, 
f{p[u.v))=Df{u.v). 
In short, in all operations in which differentiation and integration are concerned, 
D is to be regarded as if it were an ordinary constant coefficient. 
Again, if we have an equation of the form 
u.v-\-u’ .v' -^-u" .v" -\-&ec. = 0, (1.) 
there result from it 
D(^^.^^)-l-D(M'.^;')+D(M".^;")^-&c.=0 (2.) 
And, conversely, (2.) gives (1.). 
(47.) Inverse ofD. — If w be the directrix of u.v, and therefore w=D{u.v); I may 
of course, according to the true force of the index ( — 1), assert, that u .v=D~^iv. 
Thus D’ '^w comes to be an abbreviation for the words — “ the translation whose direc- 
trix is w." 
