Dr. Slevelly on the Composition of Forces. 249 



IV. We shall now proceed to prove that the direction of the 

 resultant of two forces P and Q, whose directions contain a right 

 angle and which act together on the material point A, is that of 

 the diagonal of the rectangle whose sides represent P and Q 

 respectively ; or, what comes to the same thing, if the two forces 

 P and Q which act together at A are represented in direction 

 and magnitude by the two sides of a right-angled triangle taken 

 in order (A B and B D), the hypothenuse A D (taken not in 

 order, or) acting from A to D represents the direction of their 

 resultant, as its length has been already proved to represent its 

 magnitude. 



(1) Let P = Q, and let AB represent P, and BD,=AB 



represent (by parallelism) the force equal to P which acts in the 

 direction of A C. Then, since the line A D } (the diagonal of the 

 square, or the hypothenuse of the triangle A B D T ) bisects the 

 angle B A C, it is the direction of the resultant (by Cor. of II.), 

 and its length (by III.) represents the magnitude of the result- 

 ant on the same scale as AB = BD, represent P = Q. 



(2) Let now Dj D 2 be made equal to AD T , and join AD 2 . 

 Now let a force Q, acting along A C with P still along A B, be 

 such as in magnitude to be represented by B D 2 , then shall 

 A D 2 represent, in both direction and magnitude, the resultant 

 of P and Q represented respectively by A B and B D 2 . For pro- 

 duce ADj, and from D 2 let fall on it the perpendicular D 2 B'. 

 Then, since the triangle D, B' D 2 is equiangular with A B D lf two 

 forces represented by D, B' and B' D 2 would have a resultant 

 represented by Dj D 2 . If, now, along with the two forces repre- 

 sented by A B and B D„ we introduce a force represented by 

 Dj D 2 , and along with ADj the two forces DjB' and B' D 2 , we 

 shall have the two forces A B with B D 2 , equivalent to the two 

 A B' with B' D 2 ; and therefore some one line, both in direction 

 and length, must represent the two resultants of the two pairs 

 of forces acting each together at the material point A. But 

 since the two triangles ABD 2 and AB' D 2 are equiangular, 

 and the angle at A in the one equal to the angle at D 2 in the 

 other, if we were to suppose any other line than A D 2 to be the 

 direction of the resultant of A B with B D 2 , we should be com- 

 pelled (by II.) to believe a line lying at an equal angle on 

 the other side of A D 2 to be the direction of the resultant of 



