SOLIDS ON SURFACES. 351 



of any shape and density whatever with a spherical areola of 

 contact, whirling and oscillating with a perfect rolling motion 



on an horizontal plane. The method I now oiler is intended 

 to comprise every form of this areola, having regard at the 

 same time to the nature of the surface of support. 



When the friction prevents all sliding, the elements of tin 

 curves described on the two surface! are equal, and moreover 

 coincide at every instant of the arbitrary variations, so that we 

 have necessarily (2 1) 



dx = a (5a*, -+- b dy -+- c bz . 

 dx' = a'o.r -\-b oy,-\- c'dz,, 

 dx" = a'dx'.-t-b dy,-{-c"dz r 



These values reduce ecpiations (21) to 



o = dg -\-xfia -t-y.db -hz t dc , 

 o = dp + xjba'-hyjto'-t-zfic', 



o = dg"-hx l da"-^y i db"^-z,dc"; 



or. substituting for the variations of the cosines their values as 

 given by ecpiations (6), 



o = hi -+• (c y—b z y»P h- (a z—e x,)dQ -+- (b x—a y)dJi . 

 o = ac'n- ( c 'y,— b'z,)dP + (a'z,— c'x)bQ+h'x -a y >lff , 

 o = W-h (o.y—b z )dP -4- (a z—c x )iQ ■+■ (b x —a y ),,]{ . 



• zpressions which, by means of equations (H») and the reduc- 

 tions arising from the relations (1). may be presented in this 

 form (25) 



= bi -i-z.Q — yoi;. 

 = dri -h .ri.lt — z <)P, 



o = di +y,dP—x,dQ. 



These are the relations which the condition of the peculiai 

 VOL. in. — 4 l 



