Prof. Challis on Integrating Differential 

 <H*,y,p, &c, F^ &c.)=0 becomes in the first case ^F^ 

 in the other -£__ = , and each of these vanishes if F J 



For further illustration of the eeneral th™*^ t 

 to find the differential equating 'tiT^^tST, 

 is at a given distance h from the catenarv wW? P t wI 

 tamed above. If x v k P t £ J~f al ? wh ° se equation is 

 catenary the ndrni^wtf VSSS^^.i 

 normal to the required curve at the S ? 6Ctl ° n Wlth 

 *, V, we shall hZ at tne P° mt whose coordinates 



^? 



#, y, we shall have 



or, because />, =p, "i 



Hence it follows, since ,, m c Vl+fi, that 



Consequently, by substitution in the equation of the catenary 



^^^i^T^ eqUati ° n - ^ Subs ^in, 

 /^ yvi +i? fo r ^ this^equation takes the form 



«on, or e- and c from 5 fl^T *?* the f ™ ^ 

 ferentiating to gefrid oS we ffi? ^ ^ pr ° CeSS < ^V 



or °= 1 +P 2 - W-2e ?v /I+p, 



Differentiating again to eLinate^ thtlult is 



