4 15 Mr, Landen^s LroeJligaUon of 



any where in that furface, fo that G fhould be the center of 

 gravity of the matter fo placed, any given force or forces, acting 

 on the body in the plane juft now mentioned, would caufc the 

 line NGO in the body to move exaclly in the fame manner as it 

 would move, if it were carried with the matter placed in the 

 faid furface (as berore-mentioned) after having been put in 

 motion by the aclion of the fame force or forces. Moreover, 

 let it be confidered, that there will at leail: be three permanent 

 axes of rotation in the body Q, at right angles to each other 

 (as I have proved in my Mathematical Memoirs^ ; and that, 

 fuppofmg NGO to coincide with thofe three axes in three fuc- 

 cefhve cafes wherein the matter in Q^(hall, in each cale, be con- 

 ceived to be placed in a cylindric furface as delciibed above, we 

 may conceive it poffible fo to place the matter of the body, 

 that all of it (hall be in each of thofe three furfaccs, and G 

 ftill continue its center of gravity. And, a computation being 

 iTiade accordingly, it appears, that the matter of the body Q 

 mufl: be placed^ in equal quantities, at each of the eight an- 

 gular points of a parallelopipedon (R) whole dlmenfions 

 (length, breadth, and thicknels) fliall ho, s/ zd' -\- ^f~ — ie~ ^ 

 v/2^^ -f 2/ - - 2^% and s/ ^d ^ -{■ ze^ - 2f ^ ; d, e^ and /", being 



the three values of \/GN x GO, when NGO is fucceffively a 

 permanent axis of rotation, with refpe6t to the body Q, in 

 three dlre(ftions at right angles to each other. 



If Q^were a parallelopipedon, it may be eafily prov^ed, 

 that its dlmenfions muil: be \/bd'^ + 6/ ^ — 6f% \/be' + 6/ ' — 6^\ 

 and v/6i/'4- te^ — 6/", that the correfpondlng parallelopipedon, 

 at the angular points whereof the matter of Q is conceived to 

 he placed as above, may have the fame dlmenfions as thofe 

 which we have found our parallelopipedon R mufl have. 



2 Whence 



