REPORT — 1851. 



M 



N 



The relative magnitude of the resultant is independent of the unit by which the 

 components are measured. Consequently, 



(IV.) Of two comnoneuts inclined at a given angle, the relative magnitude deter- 

 mines the relative magnitude of the resultant. 



Let two perpendicular components, repre- 

 sented in direction by M, p, be inclined to 

 their resultant S, at the same angles re- 

 spectively as two others, q, N, respectively 

 to their resultant R. 



Let M, N meet, and be equal and oppo- 

 site. It is clear by the geometry that R, S 

 is a right angle. 



R, S together are equivalent to M,p, q, N, 

 and therefore to p, q (since M, N by (II.) 

 destroy each other's effect) ; since j? and q are 

 in the same direction, p-]rq by (I.) is the 

 magnitude of the resultant of R, S. 



p-\-q is inclined to its perpendicular components at the same angles as R to its 

 perpendicular components. Hence (III.), the relative magnitude of the components, 

 must be the same in both cases. As then these two pairs of perpendicular compo- 

 nents have the same ratio, the relative magnitude (IV.) of the resultant must be the 

 same in both cases, or 



Pp+^. Similarly, ^-±i=§, .■.{p^q)-=K^+?>', 

 R 5 op 



which determines the magnitude of the resultant of two perpendicular components. 



By geometry, if the square of one side of a triangle be equal to the sum of the 

 squares of the other sides, the triangle is right-angled ; therefore if a triangle have 

 three sides representing the magnitudes of two perpendicular components and their 

 resultant respectively, it is right-angled. 



Let DOW A 6, B i represent the mag- 

 nitudes of any two perpendicular com- 

 ponents. Therefore by the above, the 

 hypotbenuse A B of the right-angled 

 triangle A i B represents the magni- 

 tude of the resultant. 



Let A e, C c represent the magni- 

 tudesof twoothercomponents, whereof -^ C 



C c=B h and A c is of any magnitude whatever. Therefore A c, the hypothenuse 

 of the right-angled triangle A Cc, is the resultant of \ c, C e. 



Let the directions of components equal to B 6 or C c meet at A and be opposite. 

 Then these components destroy each other's effect, and there remain A &, A c having 

 the same direction, and therefore Ab-\-Ac is the magnitude of the resultant of com- 

 ponents represented in magnitude by AB, AC. 



Take the straight line ca=Ab, therefore the straight line Ac=A6-l-Ac. In tri- 

 angles AB6, cC«, Ab=c\, Bb—Cc, angle A= angle c. Therefore AB=aC. Simi- 

 larly, in triangles ACc, Bio, AC=Ba. Therefore ABaC is a parallelogram. 



It is clear that AB, AC may be components of any magnitude, and inclined to each 

 other at any angles whatever. Hence, the above determines generally the maynitude 

 of the resultant of two such components. 



To find the directio'n of the result- 

 ant. 



Let AA, AC represent the direction 

 and magnitude of the components 

 of which the directions meet at A. 

 Therefore by the preceding proposi- 

 tion, A« represents the magnitude of 

 the resultant. Draw in the direction 

 opposite to that in which this resultant 

 acts, 



AE 



