106 The Rev. R. Murphy (5« tJie Composition of 



Suppose now two equal arbitrary forces, Z and — Z, to be 

 applied at the origin in opposite directions, and perpendicular 

 to the plane of X and Y: it is clear that the resultant will not 

 be influenced by this couple. 



Taking the axis of ;: in the direction of Z, the forces 

 X, Y, Z, — Z may be compounded, the first and third together, 

 giving a resultant which acts in the direction expressed by 

 the equations 



% =J^(^j . :r and y = ; 

 and then the second and fourth, which furnish the equations 



"-^=v/ ("y) '^ ^"^^ ^ " ^' 



Let the plane of these two right lines be expressed by 

 % =^ ax •\-b2/, and putting successively 3/ = x = 0, we find 



./(4) '-/(I) 



The actual resultant must be at once in this plane and in 

 that of Xi y; its equation is therefore = a x + by or 



1/ =z — ~ . X, which, compared with equation (1.), gives 



orif ^=-<. Y = # f{a)xf{^)=f{<x§), 



whatever may be a and /3 ; and it is exceedingly easy to show 

 that this equation requiresy (a) to be of the form a*": thus 

 the equation to the resultant is 



y = (x)"- ^ • • • • (2.) 



it remains to find n. 



Let d be the angle which the resultant of X and Y makes 

 with the axis of x, and ^ that which the resultant of X — Y 

 and X + Y makes with the same, the former force X — Y 

 being in the direction of .r, and X + Y in that of y ; the 

 equation to this resultant is 



y = \x^zy) • ^ • • • (^0 



