the 'Rotatory Motion of Bodies. ^ ^ 



reprefented by AB; the radius upon which ^ 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 PQjthe continuation of the great 

 circle CP. Let a denote the radius AB (=r AD) ; g and y the 

 fine and cofine of the arc CP ; s and / the fine and cofme 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 froni 

 the momentary axis ; and M the mafs or content of the paral- 

 lelopipedon (j=z^bcdj. 



Then the motive force E, urging the pole P towards Q., 

 will (by what I have proved in my Mathematical Memoirs) be 



7 ^ Di" - Qa'' ; and the motive force E, uip-ins: the fame 



pole in a dire<ftion Po, at right angles to that in which E afts, 



= 6 X T)st ; C and D being equal to c^ - if" and d"" - F re- 



II 



fpeclively. Let P^ be to Po as E to E ; complete the paral- 

 lelogram oV qr^ and draw the diagonal Pr. This laO: men- 

 tioned line will (by what I have fliewn in the FIdilofophical 

 Tranfa5lions for the year 1777) be perpendicular to tlie tan- 



I! Ill 



gent to the polar track at P. Therefore P/>/>/> being the pro- 

 je<Slion of that track, and P^ an indefinitely fmall particle 

 thereof; \i fu be perpendicular to P«A, and the quantities 



d~ — c~, c^ — b'^ be not negative; - x Dj-" — C^' will be to T>st 



(as Vq to Po) as/« toP//, the triangles Vor and Vtip being 

 fimilarv, andcr^Pj-. But with refpeft to our fpherical fur- 



face, i>u will be to P^^ as — to - — ; therefore, CV/ - Ds~ x f^ 

 ^ t 7 



will be = D^?- J J, aiid^ == — r-^^— T. Whence, by taking the 



Vol. LXXV. U u fluents, 



