286 Rev. H. Moseley on a New Principle in Statics, 



coordinates of its point of application, taken with a positive 

 sign, and subjected to the conditions imposed by the equilibrium 

 of the whole. 



Let A designate the given forces of the system, B the re- 

 sistances, and C any other system of forces which being ap- 

 plied to the same points with the forces of the system B, 

 would maintain the equilibrium. Also let the system C be 

 supposed to replace the system B. 



Now each force of the system C, under these circumstances, 

 just sustains and is equivalent to the pressure propagated to 

 its point of application by the forces of the system A ; or it is 

 equivalent to that pressure together with the pressure propa- 

 gated to its point of application by the other forces of the 

 system C. 



In the former case, it is identical with the corresponding 

 resistance of the system B. In the latter case it is greater 

 than it. 



The sum of the forces of the system B, each being consi- 

 dered a function of the coordinates of its point of application, 

 fyc. fyc, is therefore a minimum. 



Let P represent any force of the system B; oc,y,% its coor- 

 dinates ; and a, /3, y the angles it makes with the axes of these, 

 respectively. Let M v M 2 , M 3 represent the sums of the re- 

 solved parts of the forces of the system A, and N 19 N 2 , N 3 , the 

 sums of their moments ; 



.-. 3*P cos a = Mj 



^Pcos|3 = M 2 y (1) 



HP cos y = M, 



:} 



- x cos /3) = N t \ 

 — z cos a) — N 2 > (2) 



-y cosy) = N 3 ) 



XP {y cos a 

 2P (x cos y 



ZF (z cos /3 



Also «p £,» y x , a 2 , /3 2 , y 2 , &c. representing the values of 

 a, /3, y, for different points of the system B, we have 



Cos 3 a x + cos 2 P Y + cos 8 y! = n 



Cos 2 a 2 + cos 2 /3 2 + cos 2 y 2 = 1 V (3) 



&c. &c. = J 



Now the relations existing by reason of the nature of the 

 system between the coordinates of the several points of appli- 

 cation of the forces P may be expressed by the equations 



u x = 0-] 

 «* = I 



«8 = 0f 



W 



&C. SB OJ 



Also 2P = minimum = V (5) 



