• the Rotatory Motion of Bodies. g 1 9 
reprefented by A B; the radius upon which d is meafured may 
be reprefented by AD ; and the radius AC, upon which c is 
meafured, may be projected into the central point A. Let P 
/ 
be the momentary pole, and PQjihe continuation of the great 
circle CP. Let a denote the radius AB ( ~ AD) ; g and y the 
fine and cofine of the arc CP ; r and / the fine and cofine of 
the arc BQ^, to the fame radius a ; e the angular velocity of the 
body and fpherical fur face, meafured at the diftance a from 
the momentary axis ; and M the mafs or content of the paral- 
lelopipedon ( = %bcd ) . 
Then the motive force E, urging the pole P towards Q, 
will (by what I have proved in my Mathematical Memoirs ) be 
--y x Di z - Of; and the motive force E, urging the fame 
pole in a direction Ido, at right angles to that in which E a£ls, 
— a 6 x Dr/ ; C and D being equal to c 1 - b z and d z ~ b z re- 
3 a 
1 ! 
fpedively. Let Py be to Vo as E to E ; complete the paral- 
lelogram oRqr, and draw the diagonal Pr. This laft men- 
tioned line will , (by what I have fhewn in the Philofopbical 
< Tranfa 5 lions for the year 1777) be perpendicular to the tan- 
. I! Ill 
gent to the polar track at P. Therefore Pppp being the pro- 
jection of that track, and P p an indefinitely fmall particle 
thereof; if pu be perpendicular to P«A, and the quantities 
d z — c z , c z — b z , be not negative; — x D/ — Ca z will be to Dr/ 
(as P q to P o') as to Pu, the triangles P or and P up being 
fimilar, andor=Py. But with refpedt to our fpherical fur- 
• • 
face, pu will be to Pu as — to - — ; therefore, Ca z - D/ x V 
* y 
• • 
will be = Dg s s, and | ~ . Whence, by taking the 
Vql, LX XV. U u fluents, 
