OF THE TRANSLATION OP A DIRECTED MAGNITUDE. 
. 169 
II. Symbolical Representation of the two Effects produced by the Translation 
OF A Directed Magnitude. 
v. 
(20.) The Effects symbolized . — It has been shown that the translation of v along u 
is, generally, of a twofold nature, partly lateral and partly longitu- Fig- 12 . 
d\nal\ it is my objeet to express symbolically the effects produced, 
whether they are geometrical or mechanical eflFects, by the two kinds v/ 
of translation. The effect produced by the lateral part of the transla- 
tion of V along u I shall call the lateral effect, and that produced by the longitudinal 
part, the longitudinal effect. 
(21.) The two Effects are, each, Distributive Functions of u and v. Let f {u, v) 
denote the lateral effect of the translation of v along u ; Fig. 13. 
let u and v! represent the lines AB and BB' ; produce 
the arrows (v) both ways indefinitely to show the lines 
of direction in which v lies in the three parallel positions 
at A, B, and B' ; draw AGH and BG' at right angles to 
these parallel lines of direction (CD, EF, E'F'). Observe, 
that V, u and u! are not necessarily in the same plane. 
Now, as assumed, /(if, v) denotes the effect produced 
by the shifting of the line CD to the parallel position 
EF ; f{d, v) denotes that by a farther shifting, namely 
from EF to the parallel position E'F' : which two shift- 
ings ‘fut together" come to the same thing as one shifting from CD to E'F'. Now 
since AB' is represented by u-\-u’, the effect of this last-mentioned shifting is denoted 
by f{u-\-u! , v) : we have therefore 
/B 
/G '■ 
c/ 
"yd’ 
?h: 
E.' 
f{u, l^)+/(w', v)=f{U-{-ll’, v), 
that is, the lateral effect of the translation of v along u is a distributive function as 
regards u. 
In precisely the same way it may be shown, that the longitudinal effect is also a 
distributive function as regards u. For it is manifest that the three translations, 
viz. that along u, that along v! , and that along u-\-u', amount respectively to GB, G'B', 
and HB', as regards longitudinal effect, and we have HB'=GB-1-G'B'. Whence the 
conclusion is evident. 
It remains to show that both effects are distributive functions Fig. 14. 
regards v also ; and this is immediately obvious : for the trans- 
lation of V along u “ together with" that of v' along u, is the same 
thing as the translation of v-\-v' along u ; and thus, whether f de- 
note the lateral or longitudinal effect, we have 
/(w, v)-\-f{u, v')—f{u, v-\-v’). 
Both effects therefore are distributive functions with respect to v as well as u. 
MDCCCLTI. 
z 
