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



Comparing this with (3.), wc get 



/= sin ^ sin J m = — cos 6 sin i n = cos t. 



From these we find, by help of (13.), 



dflz= b sin i cos 6 -\ dp. = h sin i sin Q 

 d^l-=. — 6 sin « sin 6 J d^m = Jy^ sin i cos 6 . 



Therefore 



dH d^m 



'< " — ^t 



-b\ 



I m 



which are equations (1.) And also 



xdfl + ydp = 6 sin f (a? cos ^ + y sin 6) ; 

 that is, by help of (14.), 



which is the second of equations (11.). 

 II. Solution of the equations of motion. 

 We have then 



d;X-\-(a^^b'^)X^2bnd;Y=zO 1 



d;'Y-^(a^--b^n^)Y + 2bnd^X=0j'' 

 Let 



X=Asin(A:/ + a)"l ,^ v 



Y=A'cos(A^H-a)/ ^ *^ 



be a particular integral ; A, A', k, and a being constants. There- 

 fore substituting in the equations of motion, we have 



A\a^''bV-k^)+2bnkA=0J' ' ' ^ '> 



... {a^^b^-k^){a^''bV-k^)-4b^nU^ = 0. . (4.) 



Therefore if A', k" are the two positive values of k which verify 

 equation (4.), the general solution of (1.) will be 



X=Asin(A7 + a)4-Bsin(A'7 + ^) 1 .^. 



Y=A'co8(A/^ + «)+B'co3(F^ + /3)J * ' ' ^ '^ 



Here A' and k" are definite constants, determinable by equa- 

 tion (4.) ; A, B, a, y8 indefinite, being the arbitraries of the 

 problem. 



Also A' and B' are given in terms of A and B by the equations 



A' k^^-a^ + b^ 2bnk! 



■*■- V kf-a^-^n^^ 



5! « V-fl^ + 5^ _ 2bnk,^ _ /Jl^^ 



B "" 2bnk„ -k,^-a^-\-n b^ " -V k^^^a^^n 



A "" 2bnkf "■ kf-a'-\-n%'' 



+ ^2^2 



(6.) 



