OF THE MOTIONS OF A SOLID BO Dr. 24 J 



of gravity of the body, and being analogous to the equa- 

 tions of article 323 relatiug to a system of bodies. 



[Scholium. There can be no objection,in the strictest 

 geometrical sense, to the employment of the character d 

 to denote the element of a material body, as we have no 

 evidence to make it necessary to suppose that the particles 

 of matter are infinitely small, or that one material body is 

 ever incommensurable to anotlier : but then the particular 

 character S must always be applied to the corresponding 

 integral, which is here an actuul sum.] 



343. Theorem. Retainino- the same nota- 

 tion, (342) and making x\ y\ and z the or- 

 dinates of the particles with respect to the 

 centre of gravi4:y. we have also 



dt 



S y^£z£^ omzz Sf{Ri/- Qz') dt Bm-N". 



(B) 



Since it is necessary for the equilibrium of a sohd body, 

 that the sum of the forces parallel to x multiplied by the dis- 

 tances of their lines of direction from the axis parallel to z, 

 diminished by the sum of the forces parallel to y multiplied 

 by their distances from the same axis, should vanish : we 

 shall have 



ddy + ddi/ , , ddor + ddjr' -) 



S hx + x') Q—{i/-\-y') Plvm "(1)" : and since 



S (j:'dd?/— yddx) nmzzm {xddy — yddor), " [2]'' ; and 



S {Qx— Py) Dm=xSQDm—ySPDm, " [3]" : and lastly 



S ^(x + xO 



