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



tions of / ; and their values, as I shall show hereafter, must be 

 such as to verify the equations 



Let a^ be the central force at the unit of distance, N the 

 normal (accelerative) force of reaction of the rotating plane. We 

 have then the following equations : 



d?l_dlm_.o /I X 



I m 



n= constant = sine latitude"! .^ \ 



/2 4.^2^^2 = 1 J • • ^ -^ 

 he+my + nz^O (3.) 



£/^2y=~a2y-}-Nm I (4.) 



d^z=-a^z-^-^n \ 



Multiply the first of equations (4.) by /, the second by m, the 

 tbii'd by n, and add ; and we find (attending to equations (2.) 

 and (3.)), 



-^^Id^x^mdfy^nd^z (5.) 



Again, multiply the first by d}, the second by dp,, and add ; 

 and we have, attending to equation (2.), 



dJi.d^x-\-dfi.d^y + a^{oedjt-{-ydfi)=.0. . . : (6.) 

 Now u and v being any functions of t, we have 

 d^iuv) = ud^v + '^dp . dp + vd^u 

 = ud^v — vd^u + 2d^{vd^u) ; 

 r,ud^^v=d^^{uv)-2d^{vd^u)+vd,''u. , . . (7.) 

 In this formula write in succession, 



in place of u, /, and in place of v,a?"^ 



m, ... y >j and add; 



n, •«. ' z. 



Id^x^ f Ix- 



-i-md' 

 + nd- 

 That is, by help of equations (1.), (^.) and (5.), 



N = — 2d^(xdji + ydpi) — h\lx + my) ; 

 and agam by (3.), 



'!^=z^2d^(xd^l+ydp)'^bhlz (8.) 



"' "^ — — 



d^x^ r lx-\ r xd} 



dfy K^d^X -^-my^ — ^dA + yd pi 

 IfzJ L + w^J V-^zdp 



