Equations by Factors ana Differentiation. 391 



, jibstitutine now, as before, F for — • — — =, let it 



8 required to find the factors M and N which will make 

 iS, -f N -7— identical with the right-hand side of the equation (/3). 



u ce 



d¥ 



&-(i + p.)iv ^ y? + i+W' 



dF ,.' Nyr 



-7— which contains r is — _ 



dx (1 +/> «)1r 



d¥ . N?/r 

 e term in N -7— which contains r is — - ; and this 



f* _ 



1st be equal to the term 2 V^l+jo 2 in (/3), no other term 



(1+p 2 ) 2 

 ataining r. Consequently N = - — <£-*-> and 



MF+N^=MF- 2p yrT7 2 -^x/r+//-f-7^- 



dx qy q l vl -f-jo 



Vl+p* qy e (L+p*) 1 



?nce 

 I nnsequently the equation (j3) is equivalent to 



1 d this equation is verified if F = 0, because this supposition 



//F 

 i solves the consequence — = 0. Hence by the general theorem 



1 3 complete solution of F=0 not only gives a catenary, but 

 i ibraces all curves that may be drawn parallel to the catenary. 

 Ue.se curves, inclusive of the catenary, are plainly all involutes 

 la common evolute. 

 It is to be remarked that the integral of the equation (7), or 



