(J42 UR WALLACE ON A FUNCTIONAL EQUATION. 



dy + cdt _dx dy~cdt _ dx 



y -Vc t e' y — ct c 



and takinsi the integi-als, so that when a;=0, then y—a, and <=0, we have 



, y + c t _ X 



, y — ct X 



^ a c ' 



Hence, e heing the number which is the base of Neper's logai-ithms, 



y + c( - 

 a 



y—ct -- 



•2 = e ' . 



a 



By adding and subtracting, there is obtained 



l/=-2 { e~ + e"'^} ; (1) 



These equations, which involve in them this other equation 



f-(^fi=a\ (3) 



express the nature of the curve of equilibration. 



29. It was found (Art. 28) that (^d(ta,n(f>)=ydx=d (area BPQC). 

 Hence, by integration, putting s to denote the area BPQC, 



( = c^ tan <p 



30. By trigonometry, the subtangent QK is equal to PQ . cot PKC ; therefore. 



, , rw e' -V e ' e ' +1 

 subtan. QK- = e . = c . — . 



e' — e ' e ' —1 



In this formula, the number e = 2.7182818284. The numerator and the de- 

 nominator of the fraction — will, therefore, both increase continually with w. 



The ratio of QK to c, will, however, evidently approach to that of equality. Thus 

 it appears, that c is the value of the subtangent when x is infinite. 



31. It appears that the equation of the curve of equilibration contains two 

 constants, a and c, like those of the elhpse and hyperbola, the constants of which 

 are the semiaxes ; these enter similarly into the equation of their cmwes, but 

 here the constants do not enter similarly, for one, viz. a, enters as a coefficient, 



and the other, — , as an exponent. We have already named a the parameter of 



