VOL. LXVII.] PHILOSOPHICAL TRANSACTIONS 155 



the preceding article, and the distance c/ being denoted by v; if /; be urged di- 

 rectly from the said plane by a force fu X p, the efficacy of that force to turn 

 the said plane about the line hci, drawn in it at right angles to ocq, will (by 

 the property of the lever) be equivalent to the force-'— — ?-, acting on the said 

 line oca at right angles to the said plane at the distance g from the point c. It 

 is also obvious that, casteris paribus, the efficacy will be the same let the distance 

 of (] from / be what it will. 



In fig. 21, let 9 coincide with /: and let c^ be a line in the plane dp continued 

 (which plane will be at right angles to the plane DOEPao ;) alsc, pk being at 

 right angles to ck, let those lines pk and ck be denoted by iv and x respectively. 

 Then the sine and cosine of the angle Aco to the radius i, being respectively 



denoted by m and n, the force-'— ^? will be = - — — X {mn X (tv'' — ar^) -f 

 (ni^ — n") X u'a\) Consequently, if each particle of any solid body, through 

 which a line hci and a plain doeifqgh may be conceived to pass, be urged from 

 that plane by a force expressed hyfu X p as above ; the force which, acting on 

 the line oca at the distance g- from c, would be equivalent to the efficacy of all 

 the forces acting on the several particles of that body to turn the same about the 

 line HCI, will be obtained by computing the sum of all the forces-^- X {mn X 

 (m;2 — x") + (m' — w") X u'x) acting on the said body. 



The computation of such equivalent force will in most cases be abridged by 

 observing that, if pk be continued to p", so that kp' be = kp, the efficacy of 

 the force on the particle />", to turn the body about the line hci in opposition to 

 the force on the particle p, will be represented by the equivalent force 

 •LiLL X {mn X (-1" — w'^) + (m^ — n^) X wx) acting on the line oca at the dis- 

 tance q from c ; and that therefore the efficacy of the two forces on p and p", 

 to turn the body about hci, will be represented by the equivalent force — — ? x 



mn X {w'^ — x'^) acting on the line oca, at right angles to the plane DOEiFacH, 

 at the distance g from c. 



3. In fig. 22, 23, if the body be a cylinder, a spheroid, or the like, and its 

 proper axis be situated in the line ck, the ordinates corresponding to the absciss£e 

 kp, kp", in the circular section hi whose centre is k, will each be parallel to that 

 diameter passing through c, about which the body will be urged to turn ; and 

 each of those ordinates will be = ■/(z/^ — w^), y being the radius of such sec 

 tion. Therefore, writing 2v'(3/' — w'^) instead of />, it follows that -- X wr^a;' 

 X {^ - xY), the whole fluent of ^^"^ ^(/-^') ^^ ^„^' ^ ^^2 _ ^2^ ^ ^^ 



generated while w {■= hp=. kp") from o becomes equal to the radius y (both x 

 and y being considered as invariable) will express the value of the force which 



X 2 



