"Where a «i a^ are arbitrary constants. It may be easily 



shewn that a is the same for all values of r, and since 



12 On the Constitution of Matter, 



And since 2 cos 6 = i ^4 Y + 2 , - 4 sin^ ^ fi = 1, f^ V 

 «r \dt/ on^ \dt/ j 



the equation for determining x-i becomes 



Tfy-d^"-^' ) \df ■" f") Kdf "- ^W ^^ 



= (2) 



where ju,s = 2 »^ sin sy, s being any number from 1 to w — 1. 



If — - = when f = 0, for all values of r, we have 

 dt 



Xi=: a + aiC08[x,it + ... + fiSg cos [ji,^ t + «n-i cos ja.n_i t 



I arbitrary co: 

 or all values < 



cos (r — 1^) fl 



^' cosTi '^^ 



and that, when operating upon cos ^^, d = 2 s y 



^ = "^-1 cos(9,r — l)sy 



£Cr ^ « + 2 <^s COS /Xsi5 (3) 



^ COS 5 Y 



s = l ' 



If a?r = when ^ = 0, for all values of r, we find in the 

 same way 



s=n-i cos(2r — l)5y . ^ ... 



x,= ht + ^ I- ^ — ^ sin|U.s^ (4) 



^ cos 5 y 



8=1 ' 



r being the number of the atom, s the number of the term in 



Xr and h h^ arbitrary constants. 



Suppose the initial conditions to be 



X. = <P (r), ^J = 

 at 



where f is of any given form. For the determination of the 



arbitrary constants, we have n equations of the form 



^ = — ^ cos (2r — l)sy 



6 = 1 ' 



It may be shewn that if ^ and q be any two integers 

 2 ~ cos (2 r — 1) ^; y cos (2 r — 1) ? y = 



except when^ = q, when the sum is -. If then the equations 

 of the form (5) be multiplied respectively by cos s y, cos 3 s y 



