362 Professor Sir William Rowan Hamilton on 

 therefore 



M (l + ,)-- 1 T ffl + 1 = - n ^(l + , r »*T„. . (10 .) 

 This equation in mixed differences gives, by (9.), 



T „i (1 + y (_±_)— i l J= i. (u) 



m 4> 1.2.3... {m- 1) \d log t) 1 + *' * K } 



the factorial denominator being considered as = 1, when 

 m = 1, as well as when m = 2. If ?w > 1, we may change 



j ( 2 



^ — — - to - -, from which it only differs by a constant; and 



1 + t 1 + t J J 



h 2 



then by changing also t to e , and multiplying by — , we ob- 

 tain the formula : 



(12.) 



(e h +l) m (ay 1 - 1 . 



1.2.3... {m- 1) \dh) ( e + l ) 



2 m-l A» / s inaA w , *v* , ' . 



7T (A)l»/0 V or / v " 



which conducts to many interesting consequences. A few of 

 them shall be here mentioned. 



3. The summation indicated in the second member of this 

 formula can easily be effected in general ; but we shall here 

 consider only the two cases in which m is an odd or an even 

 whole number greater than unity, while h becomes = after 



the m — 1 differentiations of (e k -f l) -1 , which are directed 

 in the first member. 



When m is odd (and greater than one), each power, such 

 as / e h \ k in the second member, is accompanied by another, 



namely ( — e ) m ~ , which is multiplied by the cosine of the 

 same multiple of tc\ and these two powers destroy each other, 

 when added, if h — 0: we arrive therefore in this manner at 

 the known result, that 



\Th) (** + !) -1 = 0, when h = 0, if p > 0. . . (13.) 

 On the contrary, when m is even, and h m 0, the powers 



(— e l ) and ( — e l ) m ~ are equal, and their sum is double of 

 either ; and because 



(— 1) ; '{1 — 2 cos 2 # + 2 cos kx — ... 



+ (-ir- 1 2cos(2px-2x)\ = - COS ( 2 P x -"\ 



* J COS X 



by making m = 2p we arrive at this other result, which per- 

 haps is new, that (if p > and h — 0) 



