12 
of several successive rectilinear motions, or the magnitude of 
the statical sum of several forces acting together at one point, 
as a function of the amounts of those successive motions, or 
of those component forces, and of their inclinations to each 
other, we have only to denote the components by the vectors 
(yy doy » « + Gn; and their sum by a, the corresponding magni- 
tudes being Aj, Ay, ..- Any and A; and the equation 
aatataz-+-..-+ an 
will give, by being squared, 
a’ = a; + ay + ee: By 
+ apa, + aga; +... + aant ang +.--.3 
that is, by the foregoing principles (after changing all the 
signs), 
A?= A? + A.?+4+... A,’ 
+ 2A, A, cos(A;, Ao) +... + 2A; Ancos(A; An) +... 5 
a known result, it is true, but one which can scarcely be de- 
rived in any other way by so very short a process of calcula- 
tion. For it is not quite so easy, on the algebraical side of 
the question, to see that 
(Za)? + (Sy)? + (Sz)? = D(a? + y® + 2”) + 23 (aa'4 yy’ + zz’), 
however easy this may be, as it is to see that 
(Sa)? = B(a*) + S(aa’ + aa): (f) 
although the geometrical interpretation of the first of these 
two formule is of course more obvious than that of the latter, 
to those who are familiar with the method of coordinates, and 
not with the method of quaternions. 
Again, let us consider the more difficult problem of the 
composition of any number of successive rotations of a body, 
or, at first, of any one line thereof, round several successive 
axes, through any angles, small or large. Let the axis of the 
first of these rotations have the direction of the vector-unit a, 
(a* = — 1), and let the amount of the positive rotation round 
this axis be denoted by a, which letter here represents still a 
