404 Mr. Challis on the general Equations 



Then /(m - z) - F{m + z) =o, whatever be z. 

 Hence f{m) - F{m) - \f'{m) + F'(m)\z+ {/" (m) - F" [m)} | -«&c. = 0, 



independently of any relation between m and z. Therefore 



/ (m)-F {m) = 0, /' (m) + F' (m) = 0, 



/" {m) - F" (m) = O, f" (m) + F" (m) = 0, 



. /■' (m) - F'Mm) = O, /' (m) + F' (m) = 0, 

 &c. &c. 



These equations are to be satisfied so that m may have in all 

 the same arbitrary value. If we suppose / and F identical, we 



must have 



/' im) = ± f"'{m) = ± /' {m) = &c. = 0. 



These conditions are satisfied by the equation 



f'{m) = sin (w + 7), 



when ?H = — 7 an arbitrary quantity. Again, if/ and -F be 

 identical, we must have 



f{m) = ± f"{m) = ±f"{m) = &c. = 0. 



The required conditions are satisfied by the equation 



/{m) = sin {m + 7'), when m = - 7'. 



As /'(»«) = cos (m + y), we may have 



+ (w + 7') = ± yr ~(^^ + y)[' " being odd. 



n-TT 



Hence 7 — 7'=- 



2 ' 



which shews that the arbitrary values of m, which result from 

 the two hypotheses respecting / and F, are related to each, 

 other, if /'(»!) be the same in the two cases. These two are the 

 only hypotheses on which all the equations above can be 



