The Rev. R. R. Anstice on the Motion of a Free Pendulum, 381 



Again, in formula (7.) write in succession, 



in place of Uj dj,, and in place of Vyx\ , , , ^ 

 ••• dfi, ... y-i * 



/ dfd,^a^\^J ^^t^\^2d( ''^"^\ + ( ^^"^\- 

 V + dtmd^yJ * V + yd^mJ ' V + yd^^m/ \ + ydfm) ' 

 or, by help of (1.), 



\ + ydffi/ \ + my/ \ + ydffi/ 



and again by (3.), 



= d^( ^^t^\-'2bHd^-'b^( ^^t^\. 

 \-{-ydfi' \+ydfi' 



Therefore, substituting this value in equation (6.), 



d^ixdjt+ydfi) + {d^-b^)[xdj, + ydfi) '-2b^nd^z=z0. . (9.) 



Also, substituting in the last of equations (4.) the value of N 

 given by (8.), 



d^^z + {a^'~bH^)z + 2nd^{xdJ[-\- ydp) = 0. . . (10.) 



Now if, retaining the same origin, we refer the particle to 

 rectangular coordinates X and Y in the rotating plane itself, and 

 make the line of nodes t^e axis of X, I shall presently show that 

 we must have 



z^sf\-n^X 

 xdjt-\-ydfff=h */\ 



s;i} '"■ 



Substituting these values, equations (9.) and (10.) become 



J/X+(fl2-62)X-257ii,Y=0 \^ . 



d^Y-\-{a^-bV)Y-^%bnd^lL=0-^' 



which are the equations of motion in their simplest form. It 

 remains to establish equations (1.) and (11.). 



Let, then, i be the inclination of the rotating plane to the plane 

 of xy, 6 the inclination of line of nodes of said plane to axis of x. 

 Then of course i = colatitude, cosi=?i, and also 



dfi^h (13.) 



Then 



a?=X cos ^— Y sin ^ cos ? 



?/=Xsin^+Ycos^cosn ^ ^ ^ n4.) 

 ^=Ysini= \/r=^Y y 



which also is the first of equations (11.) Multiply the first by 

 sin 6 sin t, the second by ■— cos 6 sin 2, the third by cos i, and add j 



.'. a? sin ^ sin i—y cos 6 sin i +5r cos 2=0. 

 Phil Mag. S. 4. Vol. 2. No. 13. Nov. 1851. 3 D 



