14 The Astronomer Royal on the ConJilioiis 



Last term in the equation for y = 



And the equations of Article 23 become 



,^.r ,df^dx\ „/ dx\^ dx . Jd<i^ 



ttt 771 71 77i 



Dividing by 9'; remarking that —i~-^'—, — ^^o.} and that 



— r = ^^j the equations become 

 



These equations are accurate, on the suppositions that the 

 resistances are proportional to the square of the velocity, and 

 that the coefficients for lateral and for longitudinal motion are 

 different ; and they are perfectly general, as regards the form 

 assumed by the cable. 



25. AVe shall now limit our supposition of the form of the 

 cable-curve to that of the straight line, and shall ascertain the 

 conditions under which it will satisfy these equations. Let 

 ?/=y.,r,/ being a constant multiplier; s= -/(l -h/^) -x; 



dx _ 1 dy _ f f m^ \dx_ 1 /^ m^ \^ 



ds- V{\-^P)' ds~ s/{\+P)' K n^Jds'~^{l+f)\ n^")' 



^ffT-— y-^\ _ 1 dT 1 dT dx _ 1 rfT 



Ts \\ ^ ")~dsj ~ V{l+f) ' ds~ \/{\ +f) ' dx' ds~\+p' dx' 



similarly, 



dffr^ ra^,yy\- f dj 

 ds LV n" JdsJ I+/2 dx' 



Substituting these, the equations become 

 1 dT B Jm 1 



0=T-T-«--^ + 



B Jm _ I V 



l+P dx^iV \n V(l+/V '/(l+r) {l+ff 



i+r dx '^A.'\n v(i+/v ^^i^+P) (i+y^^* 



. . dT . 



2G. To eliminate -7-, multiply the first equation by /, and 



