SOLIDS ON SURFACES. 3 17 



values (6). reducing by means of equations (20), and observ- 

 iiiLr that the differential equations of the surfaces give us 



L r — L'dx -h L aV = o , Lfa ■+■ Moij -+- iV<)z, = o . 



w. shall find 



o = Lbl +L'd£ +re + ( i Vi/-i)/:),7'. 



+ (L,z t — Nx,)dQ, 



-h(Mr —Ly)nli. 



which, by virtue of the relations (10) and (11), may be also 

 rented in this form. (-22) 



o = L oi -+- Mfa -+- NX + (JV>, — Mz )bP . 



+ (L,z -N r r )„Q. 

 + (M l x-L, !/ y,I<: 



remarkable expressions, independent of the variations <>| flu 



point iif contact, and containing the required relation betw< i a 



'In variations of the station and aspect of the body made ne- 



iry by the condition of its contact with the given surface 



■ ! support. These equation* are in other respects independent 



the manner in which the body is forced to slide, roll, or 



whirl upon the surface, and are therefore true under every 



thesis of friction. 



It may he well to observe that these equations, the last foi 



mple, maybe obtained by another method which introduce - 



formulas that may be frequently useful in geometrical ;>s well 



i- in physical inquiries. If we investigate the equations (12). 



radhavi regard ins > doing to the present variability of r.i/.z. 



we shall find ( 1 i) 



, —fix, = rtC _ y ,)Jt -+_ r -Q . 



nil —nil = n K —Z n P -¥- 'II. 



- : = &—X,i <j — !/'P 

 vol. III. — I T 



