OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
185 
the combined effect of the couples will be 
2(z/ . U) = L/3 ..y + My . 05 -f-Na .|8 
=D“'(Lo5+M/3-l-Ny) (art. 48). 
Hence, if we assume G to denote the magnitude and y' the direction of La+M/34-Ny, 
which gives (art. 17) 
G'=L'+M»+N> (1.), y=g»+“/3+g5' (2.) ; 
we find 
2?LU=D-'(Gy') 
= Gci'.(5' (arts. 48 and 16). 
Here a'. (5' denotes a unit-couple in a plane at right angles to y' ; and thus the 
resultant is a couple G in this plane; G and y' being given by (1.) and (2.). 
(77-) To combine a given Set of Forces acting at given Points of a Rigid Bodij . — 
Let the forces be U, U', U," &c., and their points of application u, u', u", &c. ; then, 
by art. 74, their combined effect will be 
2(U-l-^^.U). 
Hence, if we put 
2X=X^, 2Y=Y,, 2Z=Z„ 
and employ the notation of the preceding article, t'ae ( (nubined effect becomes 
X ^05 -f- Y^)3 -f Z,y -|- Gos' . /3', 
or R^o:^-]- G o5\(3^, 
where R^=X.^-]-Y^-^Z% (3.) 
and 
ii. 

Thus it appears that the set of given forces are combined into a single force R^05j 
(given by (3.) and (4.)), and a single couple Ga'.fS' (given by (1.) and (2.) previous 
article). 
From the result just obtained the various well-known conditions and equations, 
relating to the effect of a set of forces on a rigid body, immediately follow. To ex- 
emplify the method I shall apply the formula R^cz^-j-Ga'.jS' to the following question. 
(78.) 7b combine the set of forces into Two Forces . — Let us so 
choose a' and |3' (which are arbitrary, except so far as they are per- 
pendicular to y'), that |3' shall be in the same plane as y^ and y' ; 
and let 6 be the angle which y^ makes with y'. Then, by article 17, 
y,=y' cos sin 0 ; 
.•. R^y^-)-Gc4'.j3'=(R^ cos ^)y'-l-(R^ sin 0)f3'-{-Goi' .(3' 
= (R^ cos ^)y'+(R, sin 0)f3'+ 
Now by article 74 this is the expression for the effect of two forces, viz. — 
R^ cos ^ in the direction y', 
and R^ sin ^ in the direction (3', 
2 B 
MDCCCLII. 
