030 DB WALLACE ON A FUNCTIONAL EQUATION. 



Now, when x=0, then y=fli, and since 1/=^ (""^17) ' therefore m + — = 2, 



and m' + 2 + — - = 4, and u°-2 + —r = 0, and « = 0; hence u' = I, and 



2 log u = 0, and log 6 = 0. On the whole, 



^ = log«; 



- 1 _i 



and u = e', and — = e ' , 



and y = I (« + l)=||. --.-), 



here a is the value of y when ir=0, and e is the base of Neper's system of lo- 

 garithms. 



13. We have now found that the functional equation 



/ W / W = G{f(Xo + ^.) +f K - ^0 } 

 may be satisfied in two ways, viz. by making 



f(x) = a cos — : (1) 



or 



1 1 



If we make e' = r, that is, — = Nep. logr, the second function may also 



/W = |{- + ^"~}- (2) 



r = r, tl 

 be expressed thus, 



A=^) = i{^ + >-^}-- (2) 



thus our problem (Art. 7) is completely resolved. 



APPLICATIONS OF THE FUNCTIONAL EQUATION. 



I. TO THE FUNDAMENTAL THEOREM IN STATICS. 



14. The foundation of Statics is the theorem implied in the expression, The 

 Parallelogi-am of Forces. This proposition, which in substance is due to Stevinus, 

 has been proved in three different ways. 



(1.) By the principle of virtual velocities, which is the foundation of Dyna- 

 mics. 



(2.) More legitimately from the Theory of the Lever, first estabUshed by 

 Archimedes. 



(G.) By means of a few axioms of Statics, as simple and self-evident as those 

 of Geometry. 



This last way of treating the subject was first given by Daniel Bernouilli, in 



