SOLIDS ON SURFACES. 335 



II. 



Mathematical Investigation of the Motion of Solids upon Surfaces, in 

 the Two Hypotheses of Perfect Sliding and Perfect Rolling, with a 



Particular Examination of their Small Oscillatory .Motions. 



Let us now refer, as usual, the oscillating body (.1/) to two 

 systems of coordinate axes, one of them, which 1 shall call 

 space axes, fixed in space, and originating at any fixed point 

 (O). the other called body axes, invariably connected with the 

 body and originating at any given point (O,). Let .r, x. x . 

 x. >/. z. denote the coordinates of any element Dm of the 

 body referred to these two sets of axe^: f, £', £ . £. K . (. 

 the coordinates of 0„ reckoned from 0. parallel respectively 

 to the •-pace and body axes; ./. ]}. ('. j>. ,/. ,-. the moment- 

 of inertia and the velocities of rotation round the bodj axes; 

 /•'. G, //. /'. (}. J{. the integrals Sy,z,Dm, Sz,x Dm. Sx,y Dm. 

 fpdt,fqdt, fnll: X. X . X . X. Y. X. the accelerative forces 

 in the direction of the space and bodj axes; and finally, tin 

 symhol il denoting the differentia] coefficient with respect to 

 the time /". ht </.r. thj . ,/:. ,/■ . ,l r . ,1;. ,/.,•. ,/•_,, . ,/ c . ;u ,d 



•1 1.. dr. eP£„ denote the velocities and accelerations of Dm 



and O. in the direction of the axis of the body. 



As tie- general formula of Dynamics i-. by its nature, inde- 



* [ hav.- ventured upon tin- modification of the usual notation, at tl uon 



of a valu<.l friend, principally with a new to save ruum. pmbol ■•■ 



dj- 

 ' orm d7' ' rning more or less of trouhle and delaj to the printer, 



evidently makes ev< ry line in which it k introduced take «\> more than double the 



b which it would occupy without it The Roman d will be n ei 

 m these Transactions) for simple differentials. 

 VOL. in. — 4 q 



