COMPOSITION OR AGGREGATION OF FORCES IS A CONSEQUENCE. 293 



8. If the aggregants be altered in any ratio, without change of application, the aggregate 

 is altered in the same ratio, also without change of application. Let A^B be C: then, m 

 being any integer, {mA) ^ (mB), where m affects only the magnitude, is by 4, A^A^A^... 

 xfixfix^x... or, by 3, A"" B"" A"" B^" . . . , or, by 5, C'C'C''..., or, by 4, mC. Again, if 



l-^Yi- ^) give F, A><B is nP, or C is nP; whence P is i C. Hence (- aY (- b] 



m 



is — C. Hence it follows that the ratios of the aggregants to the aggregate, and the relative 

 n 



application of the aggregate, depend solely on the ratio and relative applications of the 



aggregants. 



9. Any tendency may be disaggregated into two of any two different applications, 

 neither of which is its own : that this cannot be done in more than one way has been proved. 

 Take any tendency of one of these applications ; then, by taking another of sufficient smallness 

 of the other application, an aggregate may be produced which shall be as nearly of the first 

 application as we please. Again, since the relative application of the aggregate depends only 

 on the ratio of the aggregants, by taking the second aggregant as great as we please, we may 

 produce an aggregate as nearly of the second application as we please. Consequently, take 

 what tendency of the first application we may, we can find another tendency of the second 

 application such that the aggregate shall be of the same application as the tendency which it 

 is required to disaggregate : and, by 8, alteration of these two aggregants in the proper ratio 

 will give the aggregate the proper magnitude. In assuming tendencies to have magnitude, 

 the law of continuity has also been assumed ; which is mentioned here because this is the first 

 place in which the assumption has taken effect. 



We are now prepared to deduce the modes of aggregation when specific matter is added 

 to the forms discussed above. Two cases arise : first, when by application we mean direction, 

 from which we deduce the diagonal law of aggregation ; secondly, when by application we 

 mean choice of a point at which to apply a given direction, from which we deduce that law 

 of inverse ratio which is commonly called the principle of the lever. 



]. For the first case, any one of what are called proofs of xhe parallelogram of forces may 

 be inserted. The whole proposition follows, in well known ways, so soon as it is proved for 

 tendencies at right angles to one another. Let P and Q be two tendencies at right angles to 

 one another, and let R be the aggregate, making an angle 9 with P. Then, as shown, there 

 exists such an equation as P = It (p9. Let two equal tendencies, of magnitude P, be inclined 

 to a certain axis in their plane at angles a; + y and x — y : then, from 2, the aggregate of P 

 and P is inclined to this axis at the angle x, and to each of the aggregants at the angle y. 

 Let P and P be disaggregated in the directions of their aggregate and its perpendicular : the 

 perpendicular aggregants counteract each other, and each of the others is Pcpy ; whence the 

 aggregate is 2P<py. Disaggregate the two tendencies and their aggregate in the directions of 

 the axis and its perpendicular : the aggregants in the direction of the axis are P (j) {,v + y), 

 P(b{cV-y), and iP(piv(py; whence 



(p {oD + y) + (p (x - y) = 2 cpx (j>y. 



