By Professor Pell. 15 



where as before 9 = 2 + — I --I = 2 cos 9 



^ m^ \dt/ 



If these equations be multiplied \)J fi fi respectively, 



and added together, and all the terms disappear from the 

 resulting equation, except those involving x^ and x^, we have 



2 cos 3 /,=/,_! +/ + ! 

 which gives j/^ = ^ sin (r 9 + B). 



The condition 2f^ cos ^ =fi, gives 5 ::= 0, so supposing^ = 1, 

 we have 



^ sin r 9 



Jr — ; 7- 



sm fl 

 and the corfficient of x^ is 



(2 cos a - 1)/. -/._,= ^^Ii-JL^)J 



cos i 9 



The equation is therefore 



cos (n + \)^ , 



}: -<_ x^ = a cos u, f 



cos i 9 



We have also 



x^ = A cos (r 9 + jB) 



and the condition (2 cos 9 — 1) x^ = a^n-i gives 



B = — ^r — 9, so that 



tCj. = ^ cos (^i — • ^ + i) 9 



£Cn ^ ^ cos i 9 



^ ^ cos (^^ — r + i) 9 ^^ 



cos i 9 



cos (n — r + \') ^ . 



= a i ' cos \i.i 



cos (w + i) 9 



If -we put |W, = 2 m sin 4/, then, when operating upon 

 coBat, 9 = 2 4/, and 



cos (2 n ■ — 2 r + 1) ^ 



cos (2 w + 1) vj/ 



COSjU,^ 



The tendency to rupture is a function, not of the displace- 

 ments, but of the relative displacements of the atoms, represented 

 by 



:,^ ^^_^_ 2asin(2^-2r + 2)^^sin^^ ^^^^,, 



cos (2 ?i + 1) 4* 



