332 Dr. Hamilton on the Plants of various Parts of India, 

 the 7iatural integral when ?i is a whole number ; and 



A-\ = (21) + a,A^-".x° + a,A'-"..rO +...«,,_, A^-V^22) 

 the complete integral n being a whole number. 



Should A w = — '■ we shall have 



for the natural integral when n is a whole number and the 

 common difference = a. And because 



if in (23) we put a = we find 



» n+I "+2 



rf-\ =-^— w +^,^— dti + 3„ .^_^l!f_... (25) 



■^ 1.2...n X "l.2...(»+l) ^ 1.2...(ni2) ^ ' 



which coincides with (15) when r-=n = a whole number. 



We see from (24) and (25) that the calculus of differences, 

 and the differential calculus are not nierely similar, but one 

 and the same in principle ; the differences in the one case be- 

 ing finite and in the other nothing. Hence the reason of the 

 strong similarity in the processes, though it should be ob- 

 served that what is usually imderstood by the method of dif- 

 ferences is in fact the calculus of functions. 



It would be supei'fluous to make any observations on the re- 

 markable simplicity and generality of these formula?. Nor 

 can it be expected that I should here notice the variety of 

 purposes to which the double liability of (21) and (23) to ter- 

 minate may be applied. These speculations may probably 

 form a part of what I may have to say of the calculus to which 

 I have already alluded. 



LIV. Some Notices coticerning the Plants of various Parts of 

 India ; and concerning the Sanscrita Names of those Regions. 

 By Francis Hamilton, M.D. F.R.S. Sf F.A.S. Lond. and 

 Edin.* 



AS it is my intention soon to publish, in various works on 

 Natural History, the observations on the Botany of India 

 which I made during my residence there, I wish to place on 

 record an account of the opportunities which I enjoyed of 

 making such observations, with the view of exjilaining to the 

 botanist where he may find the various collections which I 



* From the Eiliii. Pliil, Trans, vol. x. part I. 



made 



