32 2 Mr. Landen*s Invejllgation of 



ab (or a circle) whofe center is A ; femi-axis A a=;z ; and the 



other femi- axis = — i x 7z ; excepts be = ^; in which cafe the 



proje6lecl track will be a right line a b parallel to AC. 



With regard to the permanent axes of rotation of our paral- 

 lel opipedon, it appears, by my Mathematical Memoirs^ that if 

 two of its dimeniio-ns be equal (that is, when the body is a 

 fquare prifmj^ any line paffing through the center of gravity of 

 the body, in a plane to which the other dimenlion is perpendi- 

 cular, will be a permanent axis of rotation ; as will the line 

 paffing through that center, at right angles to that plane. If 

 all the three dimenlions be equal (that is, when the body is a 

 cube), zny line whatever paffing through the center af gravity 

 of the body will be a permanent axis of rotation. 



It is obfervabLe, that the momentum of rotation of the 

 body, about the momentary axis, is found by computatiork- 



always = -V X b'm~ -f ca' + d^^n\ e denoting the angular velocity,- 

 But 4- X ^'^^"^ +c'a~ -{-d^n'' is the initial momentum of rotation. 



a^ 



Therefore, confidering the momentum of rotation as invariable, 

 the angular velocity will be invariable, e being always =yi 

 which here denotes the initial angular velocity. 



Our next bufinefs is to find the length of the track defcribed 

 by the momentary pole (P), upon the fpherical furface; and_ 

 the velocity of the pole in that track. 



Fig. 2, 3. It appearing, that the motive force E is = 



-4" X D//r - Ca~ X - , and the motive force E - -4- ^ y^^^'^~ - ^y 



X v/0//-B7' ; we fnd F = V E^ + E' (the force compounded 



of thofe two forces) = -r ^ v/D'w V - BC^^y ; and, F being 



1 to 



