and on Fourier's Theorem. 487 



Q b + 2cx 

 * — Sx^ 



-3j;2 



■ _ Z> + 2g~{(3w + l)6 + (3n + 2)c+(3?z + 3)a} 



3 

 = a\ 

 and, similarly, 



Sg = flr + i, S3 = « + ^ + c; 



and this quite independently of «, whence it is clear that the 

 proposed series has three values, «, a + b, a-{-b + c, and is not 

 equivalent to the mean of those quantities. 



It is plain, therefore, that this supposed illustration does 

 not bear out the principle it is adduced to support. 



Mr. De Morgan proceeds to say, "2. In applying Fou- 

 rier's theorem (p. 629) to discontinuous functions, we find 

 that at the point where the discontinuity takes place, and a 

 function which generally can have but one value might be ex- 

 pected to take two, it takes neither, and gives only the mean 

 between them." 



It will take some trouble to unravel the maze of errors in- 

 volved in the above short sentence. 



Recurring to p. 629 of Mr. De Morgan's treatise, we find 

 him there proposing to find, by means of Fourier's theorem, 

 "a function of .r which is = ^ from ,r = to a? = 1 and = 

 everywhere else;" and he actually finds a function which he 

 believes (most mistakenly) to satisfy the conditions in gene- 

 ral, but which at the limit when ^=1, instead of giving a?= 1, 



or a; = 0, or both these values, gives x = — . 



Before entering on the question of the analytical errors 

 upon which this argument is based, I would say a few words 

 upon its irrelevance to the subject in hand ; and with this view 

 I would observe, that there is no reason whatever to expect 

 that in this case, when x = 1, the function should have two 

 values. The conditions of the problem are, that the func- 

 tion shall be =a; from ,2? = to a; = 1, and =0 everywhere 

 else. Because when ;r = 1 the function is to be = 1 , and 

 when X is greater than 1 it is to be = it is rather an ex- 

 traordinary conclusion, that when .r = 1, the function should 

 be 1 and una voce; yet this Mr. De Morgan thinks " might 

 be expected." But to cut this matter short I shall now show 

 that the principles upon which Mr. De Morgan interprets his 



