The Rer. E-. R. Anstice on the Motion of a Free Pendulum, 383 



The problem will be much simplified if we suppose the period 

 of oscillation of the body very small compared with the period 

 of rotation of the plane. Then b will be very small compared 

 with a, so that its square^ &c. may be neglected^ and our equa- 

 tions become 



,\k^=:a-\-hn k! =^a—bn. 

 Also ^'«.i S'_ _ 



.*. X=Asin («^ + a + 5w^) + B sin {at+^—bnt) \ . . 



Y=sAcos {at + u-{-bnt)-'Bcos{at + 13— but) J ' 



Now were the term bnt involved in these equations constant 

 instead of a function of the time^ the orbit we know would be 

 an ellipse round the centre, or a straight line. That term, how- 

 ever, contains tj and will in process of time become sensible ; 

 but as it alters with extreme slowness, we may consider it as 

 sensibly constant during one oscillation of the body, and deter- 

 mine the elements of the ellipse on that hypothesis. To do this, 

 consider for a moment the equations 



a; = CBm{at-\-6)\ 



y='D cos{at + €)J 

 C, D, a, and e being constants ; 



and the orbit is in this case an ellipse, whose axes coincide with 

 those of the coordinates, and = 2C, 2D respectively. 



But if these axes, instead of coinciding with, were inclined at 

 an angle cj) to the coordinate axes, and X and Y are the coordi- 

 nates in that case, we have 



X=a7 cos </>—?/ sin ^ = Csin {at + e) cos<^— Dcos(«^ + €)sin<^ 



Y=^sin <^ + 2/cos ^ = Csin (fl^ + 6)sin (^ + Dcos(«^ + €)cos(j5>. 



That is, 



n 



X= — (sin (a^ 4- € + </)) 4" sin (a^ + e — 0)) 



+ K-(— sin(«/ + e + ^) + sin(fl!^ + €— <^))^ 



Y=— (— cos(«if + e + 0)+cos(«^-f €— ^)) 



+ g- (cos {at + € + (!>)+ cos {at+€— <^)) J 

 2P2 



