(^42 ^^ WALLACE ON A FUNCTIONAL EQUATION. 



dy + cdt _dx dy — cdt_ dx 



y + ct c ' y — ct c 



and taking the integrals, so that when x=0, then y=a, and t=0, we have 



, V+ C t X 



^og- =— , 



a c 



. 1/ — Ct X 



^ a c ' 



Hence, e being the number which is the base of Neper's logarithms, 



y + ct - 



= e'; 



a 



y-ct -- 

 a 



By adding and subtracting, there is obtained 



-J r J- 



y=^ { e~ + e~^} ; (1) 



'=^.('^--'^"') (2) 



These equations, which involve in them this other equation 



y_c2;2=a2, (3) 



express the nature of the curve of equilibration. 



29. It was found (Art. 28) that c^ d (tan (p)=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. 



3x 



subtan. Q,K = c . =c . 



2x 



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 x. 



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

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



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

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

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



