154 PROFESSOR TAIT ON THE APPLICATION OF HAMILTON'S 



10. As an example of a tortuous curve we take the following : 

 Determine the form of the brachistochrone when the velocity at any 'point of 

 space is proportional to the distance from a given line. 



Taking the line as the axis of z, our equation obviously becomes 



\dx) + \dy) + \dz) - x* + 



*rtf 

 Hence 



dr 



dz~ a > 



and, substituting this, and changing to polar co-ordinates in a plane parallel to 

 xy, 



\dr) + r 2 \dd) ~ r*~ a ' 



Hence we may take 



dd~ p ' 



and there remains 



Integrating, we have 



r = az + 3d- Vtf^p* log. \ y^^ + J'^fi- - a 2 ] + \/a a -^-oV . 



By equating to constants the partial differential coefficients of t with respect to 

 a and (3, we obtain the two equations of the brachistochrone 



en ar ' Z 



%L = Z — 



y/a 2 - fi 2 + Va 2 -B 2 - a 2 r 2 ' 



The former of these is the equation of a sphere, as may be seen at once by 

 putting it in the form 



a (z-®) = Ja 2 ^- J a 2 -B 2 -a 2 r 2 . 

 The remaining equation, by altering the value of 13, may be reduced to the form 





a 2 -B 2 ( 

 a 



2 vg -P=rU ' ' + € 



which is at once recognised as a cylinder, whose base is one of Cotes' Spirals. 

 Also, if we remark that, by (1), 



ctf_ , dr _ r 2 §__§v 

 dt rdd ~~ a 2 r ~ a 



