728 liP.IJ. SYSII-.M TECHNICAL JOURNAL 



Theorem VI 



If h=h{l) ami g = g(l) arc defined by the operational equations 



ii = \:ii{p) 

 g = \;ii{\p) 



where X is a positive real parameter, then 

 g(t)=h{l/\). 

 We start with the integral ctiuaiions 



pll(p) 

 1 



hrf"^''^-"''^ 



pll{\p) Jo ^'- 



and in the first of these equations we replace p by \q and / by t/X, 

 whence it becomes 



m=r"(iy-'''' 



qH{\q) 

 Now replacing the symbols q and - b\' p ami / respectively, we have 



pim)=r'"''^'-"" 



whence by comparison with the integral equation in g(t) it follows 

 at once that 



gW=/'('/x). 



This theorem is often useful in making a convenient change in the 

 time scale and eliminating superfluous constants. 



Theorem VII 



If h=h{l) and g=g{t) are defined by the operational equations 



^~iiiP) 



where X is a positive real quantity, then 

 g{l)=o for t<\ 



= h{l-\) for /^X. 



