Calculus of Finite Differences. 

 = j-^ | A . (» - 1 ) (n - 2) . . (n - t + 1) 



+ ^.(«-8)(«-S)...(»-/) + C ^ (/ ~ ] V -3) 

 (» — 4)...(7*— /— 1)H 5 _ — i — — '(w — 4)... 



133 



(m-*-2)&c. &c.T 



?* . n— 4. w — 5 



£ (VII.) 



Now the part of this series within the brackets is manifestly 

 the same as (V.); comparing then the two series, term by 

 term, we find 



a -ir> ^ n.n — 3 



A = 1, B = — w, 



C = 



1 2 



„ n.n — 4. rc— 5 ^ /& . w— 5 . w— 6 . w — 7 a 



I 2 3 

 and the coefficient of z n ~ is 



12 34 



+ n(n-t-l)(n-t- 2)...(»-2 /+ 1) ^ _ (VIII.) 



When 2 £ = «, the coefficient of s° becomes + 2, as was shown 

 above. 



When n is odd, the coefficient of * is found by putting 

 %t =» ft — 1 in the general formula (VIII.), when it becomes 

 ±n. 



We have now shown that 



a? + a? = z — ft-? + — — - — z ...&c. . (IX.) 



1.2 v 



or, as it may be more compendiously written, 



L ~~ \ 12 S..Jx Mt J J 



Attributing to £ all integer values from unity up to — in- 

 clusive. 



To develope the cosine ofnQ in descending powers of cos 0. 



Assume 2 cos 3 = x + x~ , then it may be easily shown 

 that 2 cos 11 6 = *"+#"" Substituting in (IX.), 2 cos w 6 

 for a/'-f- x~'\ 2 cos 9 for z, and dividing the whole equation by 

 2", we find 



