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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 

 oi, a-i, . . . an, and their sum by a, the corresponding magni- 

 tudes being Aj, A2, . . . A„, and A; and the equation 



a =: ai -{• 0-2 -{■■•■ + Un 

 ■will give, by being squared, 



a^ = ai^ + a.2^ + ... + aj 

 + ai a2 -|- a2 ai + • • • + «i an + «n «! + . . . ; 

 that is, by the foregoing principles (after changing all the 

 signs), 



A^ = Ai^ + Aa^ + • • . A„== 

 + 2 Ai A2 cos (A„ A2) -f . . . + 2 Ai A„ cos (A, A„) + . . . ; 



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 



{•2xy + {-Eyf + (^zy = ^(x^ + y'' + z-") + 2^[xx'-\- yy' -H zz'), 

 however easy this may be, as it is to see that 



(Sa)2 - S(a'0 + ^{aa' + a' a) : (0 



although the geometrical interpretation of the first of these 

 two formulae 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^ ■=. — \), and let the amount of the positive rotation round 

 this axis be denoted by a, which letter here represents still a 



