Resolution of Forces. 259 



the angle formed by their directions ; and half that resultant 

 is defined to be the * resolvent' of either of the equal forces. 



4. When any number of forces act on a point, if any 

 straight line be drawn through that point, and planes be 

 drawn through this line and the respective directions of the 

 forces ; and if in each plane a force equal to the force in that 

 plane be applied to the point, making an angle with the 

 straight line equal to that which the other force makes, but 

 on the contrary side of the line, such a system of forces is 

 called ' supplementary ' to the former. 



5. A system of forces acting on a point is in equilibrium 

 when their resultant is zero. 



Prop. I. When a system of forces is in equilibrium, the 

 sum of the resolvents in the direction of any straight line 

 through their point of application is zero. 



The system being in equilibrium, the supplementary system 

 will also be in equilibrium ; for this latter system is nothing 

 more than the former turned through 180° round the as- 

 sumed line. 



The two systems are therefore also in equilibrium, and con- 

 sequently the resultant zero. But this resultant consists of 

 the (algebraic) sum of the several resultants of the pairs of 

 equal forces in the several planes through the assumed line, 

 and half the resultant is therefore the sum of the resolvents of 

 the given system. The resultant being zero, the sum of the 

 resolvents is also zero. 



Prop. II. Denoting the resolvent of a force P in the di- 

 rection of a line making an angle with the direction of the 

 force by P/0, it is required to assign the form of/0. 



It is an immediate consequence of our definition of resol- 

 vent that /0 = 1 when = 0, and that/0 = when = -. 



Moreover, that/0 cannot = for any value of <: — . 



Also that/0 is a periodical function going through all its 

 values, as increases from to 2w; and that/0 can never be 

 infinite, and must have some determinate single value for every 

 value of 0, and is therefore a function which can be expanded 

 in integral positive powers of 0. 



Let two equal forces keep a point in equilibrium. Sum of 

 resolvents = gives 



P/0+P/(tt + 0) = O, 

 or /0+/(7r + 0) = O. 



This equation is easily resolved, but is too general for our 

 present purpose. 



