38 Statics. 



AV+BH+CKf 



and which will pass at a distance DD from FC', equal to the 

 expression below, namely, 



jy D _ AV x AA + BH x BB' + CK x CC 1 

 J1V + BH + CK 



In like manner, the forces AI, BR, CM, parallel to FB", are 

 reduced to a single one DO, parallel to AI, &c., and equal to 

 AI -\-BR CM, and which (by supposing that D is the point 

 where the direction of this force meets that of the force ND) will 

 pass at a distance D'D from FB", equal to the following express- 

 ion, namely, 



AI X AA " + BR * EW CM x CC" 



AI + BR CM 



This being supposed, the forces p, 7, r, and their directions, 

 (that is, the angles which they make with the known iixed lines 

 FC', FB", or with their parallels,) being considered as known, we 

 know in each of the triangles AEL BGR, CLK, the hypothenuse 

 and the angles. It will accordingly be easy to determine the 

 Trig. 30. lines AI, BR, KL, or CM, and the lines IE or AV, RG or BH, 

 and CK. We shall consequently know the values of the two 

 resultants AV + BH + CK, and AI + BR CM. Moreover, 

 as we cannot but know the distances A A, A A", BB', BB", &c., 

 since the position of the points A, B, &c., where the forces are 

 applied arc supposed to be given, we are acquainted with all the 

 quantities which enter into the expression of the distances D'D, 

 Jy'D. It will be easy, therefore, to determine the point D, where 

 these two resultants meet. Accordingly, taking 



DO - AI + BR CM, and-DJV" = AV + BH + CK, 

 and forming the parallelogram DNTO, we shall have the diago- 

 nal D T for the resultant g, of the two partial resultants, parallel 



* We must not lose sight of what was said art. 39. By the forces AE, 

 BG, &c., we are to understand that the lines AE, BG, &c., are to 

 each other as the quantities of motion capable of being produced by 

 the forces p, q, &c., in the masses to which they are applied. It is 

 to be observed, likewise, with respect to the forces AV, BH, &c., 

 that we mean by them quantities of motion, which are to the quanti- 

 ties of motion represented by AE, BG, &c., as AV, BH, &c. ? are to, 

 AS, BG, &c., respectively. 



