and on Fourier's Theorem. 489 



But in order that Mr. De Morgan may not be able to bol- 

 ster up his verification of his function by the supposed author- 

 ity of Fourier's theorem, by which it was originally suggested, 

 it is necessary that I should show the fallacy of the theorem 

 itself. 



The truth of that theorem depends on the proposition 



(vide Treatise on Differential and Integral Calculus, p. 616 

 ei seq.), which in its turn depends on the fact that the limit of 

 the series 



~ + cosy (^—y) A + cos-^(a;— y) A^ + &c. is 



1 1 -A^ 



^ 1 -2Acosy(a;-u) + 62 



when A = 1, which is not the case, for putting -j- (.2?— u) = d, 



If 



we have 



1 I— A^ _ 1-Acose 1 



2 1— 2Acosfl + A2~ 1— 2Acos9 + A^ 2 



2 ^ ^1 



l-A(s^^-i + 6-^^-i) + A2 2 



-L l-A8^^^^+l~A6-^^~i 1 

 "" 2'(i-A6^'^^^)(l-A6-''^^) 2 



^±{ 1 _ , ' \ ' 



^Vl-As^^-l l-Ae-^^-V 2 



■^ ri-f-As^^^ + A2e2'^^^+&c.+A»-ls»-l^^^ 



/ ,v / — Aw„— w^V'— 1 



+ &C. + A'*- ^e-('»- 1)^^-1 4- — 



l-As-''^^^ 



= ■2" + Acos9 + A^cos 2 9 ■{■ &c. + A«-icos(w-l) 9 



cos n g — A cos (?? — 1)9 

 "^ ^ * l-2Acosd + A2 



