498 Mr. Denham Smith on Ferric Acid. 



have 



/ — r(sin^r r+I 



*y x n cos.r 



?2P(»--])t 2 + 2 2 " ,+ 2 2r J. 



When * is even, xj^ 1 = - 2 w~*~ 2 + 2 t^ -4 _ & c . con- 

 tinued till they become zero, for all the terms preceding de- 

 stroy one another, and the remainder of the series supposed 

 continued to infinity is nothing; for we may prove as before 



pdx sin p a; . , . . . . „ . 



thaty — (sin x) m — £ — is nothing when p is infinite and 



odd. Hence by substitution from (c), 

 ^+i = (- 1 ) 1 * {A(?,i-z-2) n - 1 -A ] {m-z-4) n - 1 + ...} 



"m,n 2 m_1 P(w— 1) 



\n— 1 



•(G.) 



+iip(^Ij A " ( — ): 



The last term in (G.), or the correction, is to be left out when 

 the number of terms in the preceding series is even. As an 

 example, 



r~ ■ v2r + i sin (2r— l)o? _ ( — ])»• y 



J ^ {SmX) cosa? ~ 2 2 '■P(^^-l) , ' ( ' 



2r+l>rc — 1. 



We cannot employ any of the formulas of this paper if n — 1 



and O = tan -1 \~k) enter 5 because this quantity is indeter- 

 minate. In some cases O = — leads to true results, but 



perhaps it cannot be proved that it must always take this value. 

 Gunthwaite Half, March 10, 1844. 



LXX. Note on a paper on Ferric Acid, read May 16, 1843. 



By J. Denham Smith, Esq.* 

 HPHE paper above referred to, which I had the honour of 

 •*• laying before the Society last sessionf, was unfortunately 

 printed before I had proved that two material errors were 

 contained in it. These errors arose partly from the almost 

 invariable presence of manganese in the oxide of iron, preci- 

 pitated from the sulphate, which I employed, — an impurity I 

 neither suspected nor guarded against, and which usually oc- 



* Communicated by the Chemical Society ; having been read December 

 4, 1843. 



f See Memoirs, vol. i. p. 240. [or Phil. Mag. S. 3. vol. xxiii.] 



