256 



We substitute 



Mr. L. A. Par 



u(ctanhu) = 6(u) 



ky(c tanh u) — (j>(u) 

 tank -1 u/c = x 

 I tanh _1 ?;/c=y, 

 and the equations (16) become 



6(.v+y) = dUv)e(y) + 4,(.v)<j>(y) 



and since the range to c for u corresponds to the range 

 to qo for x, these are to hold for all real values of x and y. 

 But the equations (17) give 



(17) 



Q( x+y ^-$( x +y)={0(x)-<t>(x)\{0(y)-<f>(y)\, 



(18) 



so that both the functions \ {#) + (j> (x) } and {0{x) — <f)(x) } 

 satisfy the functional equation 



It follows that 



■f(x-\-y)=ylr{x)yjr(y). 



0{x)+<f>(x)=e ax 

 0\x)-cf>(x)=e dx , 



where a and b are constants to be determined. 

 We have therefore 



\x)=±(e az + e hx ) 



bx\ 



or 



(j>(x)=±(e ax —e> 



where f is written foi 



1+m/ 



for the sake of brevity. 



1 — u/c 



Moreover, a{u) is to be an even function and y(u) an odd 

 function, and this requires a + b = 0. !So w T e have finally 





(19) 



