34 Prof. Challis on a new Integration of Differential 



First, it is to be remarked that if, by the process employed 

 above, we had deduced the equation of an evolute from A^ = 0, 

 we should equally have been conducted to the equation (a), 

 because the factor p would have made no difference. But in this 

 case, since Ap = is at once integrable, we know that there is 

 but one involute, namely the catenary whose equation, as obtained 

 by the integration of y=cVl +jo 2 , is 



x+c >=chg(z+\/£-iy 



By substituting Vl +p 2 for -, this equation takes the form 



3? + c'=c log ( vl-\-p^-\-p). 

 Now, on comparing this result with (/3), it will be seen that the 



kp 



two equations are identical if ~=c and —py-\- -—■ i^l+p 2 = 0. 



But these equalities give at once the foregoing equation 



y = c\/l+p 2 ; 



which shows that the integral of Ap = is one of the evolutes 

 embraced by the equation (j3). This being the case, it is easy 

 to deduce from that integral a differential equation embracing all 

 the evolutes ; which, if our reasoning be good, ought to coincide 

 with the equation (13). I proceed to verify this conclusion. 



Let x p y { be the coordinates of any point of the catenary 

 whose equation, as obtained by integrating Ap = } is 



*, + c' = C logg+y J -l) = c log(Vn^?+i>,). 



Then if x, y be the coordinates of the point of intersection of 

 any other involute by the radius of curvature of the catenary at 

 the point x^^ or by its prolongation, and if h be the constant 

 interval between the two curves, we shall have 



hp, . , h 



Also, since y =c\/l +p*, it follows that h=c(l +/> y 2 ) —y V\ +pf. 

 Hence 



Xi + c > =os + c<- ^> (c(l ±pfi -y ST+pJ) 

 = os + c — cp, V 1 +p? +p,y. 



