574 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



the above form is applicable in all cases, and may therefore be used as the general 

 form, without any reference whatever to the specific values of n and /x. As this 

 proposition is of the utmost importance, we shall give more than one proof of it. 

 We must, however, make a slight change in the notation, in order to avoid the 

 charge of misapplying Legendre's functions. We shall, therefore, write > in- 

 stead of /" {p) or rp, lengthening the top of the /~ and assuming the function 

 I'p to be such, that, like Legendre's function, it satisfies the condition \+p= 

 P rp . By this hypothesis, /~ and / become coincident, whenever the quanti- 

 ties under them are positive. 



We suppose, then, that the general expression for 



^_^ IS 



whether w or w + M be positive or negative. 

 This we call formula (I). 



6. II. To find the differential coefficient of x"" where n is greater than fi. 

 The result will 1)e 



rf'^.a:" 1 . -.x^ /fi-n 



(-1> 



Now, /l-{n—ix) ^-{n-jj) / -(n-fj.) 



/2-(n-fJL) =+(l-n-fx) 'l-(n-fJL' 



lr—{n — )S) = +(r — l — n — lJL) /(r — 1) — (« — /z) 



therefore, by multiplication, 



/>•— (w — />() =( — 1)'' I// — ju ■// — // — 1) ... (?l — /x — /• — 1) X /—(«—/>(' 



/• being the integer next greater than n-fx. 



In the same manner, /^ may be reduced by the formula 



/s~n = ( — iy n(n — l) ■■■ (n — s — \ —n 



where s is the integer next greater than n. 

 Hence, by division, 



/ii^iT/,) = (_i). i'.-};^~^^^^ 



— I s—n 



« (w — 1) • • . (n — s + \^ 



/ ~(n-fX) ^ /r-(n-tx) « («-l) ...(»-*• + 1) 



/~n /s-n ^n-^) ... (n- fx- r+l: 



/s—n /n~s + l /n — fJL + l 



