380 ON THE MOTION OF 



of W = o, make up eight in all, being two less than when 

 the body was not restricted to the peculiar motion to which 

 we now suppose it to be subject. 



Lastly, let F and G be very small. The equation W = o 

 will now give us, as if there were no friction, r = any arbi- 

 trary constant, and R = rt-\- R. At the same time we have 



p = ra" H- db", dl = dl — m, + lq , d% = d% — ftfr, . 

 q = r6" — f/a", (7)7, = c?>7, -t- r£ — ^ . rf>, = d\, -+- rd£. , 



and, by the equations of perfect rolling, 



dt = ry,—yq, 

 dyj, =: — rx, -J- yp ; 



whence 



d% = r*x, -\-rdy, -\-y(d 2 a"-\-2rdb" — rV), 

 d\, = r\j, — rcfc, -+-y(d*b" — 2rt/a" — r*&") . 



By comparing the two values above given for each of the 

 quantities <I|, and dr;,, we obtain, after replacing %-j-y by a. 



^(a:, 4- £ ) = a/> — rf>?, , 

 r(y, -+->,) = aq-\-di . 



When the supporting areola is spherical, these become 



rTiX, = am (ra" -+- c?6") — dy , 



where a, and u are abridgments for J-+-- and m-h '-£. 



By means of the preceding values of d% and d\ it will 

 be seen that the first and second equations of motion are 

 transformed into equations of the second order involving 

 a", b", a?„ y and / with constant coefficients. These, in con- 



