S22 Mr. J. Herapath on the Binomial Theorem, 



October 17, 1822, informs me, that he has long had an idea 

 of a "new calculus" by which " we may inquire what is the 

 nature of an operation P to be performed / times in succes- 

 sion (according to its own peculiar rules) on any function ipx 

 which shall produce fi-om it 



<p(x+1) -4>jc;" 



and of course give P' ^ .r = A<px and P = A ' . This, how- 

 ever, is confined to the method of differences and rational 

 orders, and will not do for irrational, nor, as Mr. Herschel 

 observes, for imaginary orders. What length Mr. Herschel 

 had then or has subsequently carried his researches I have 

 not heard. From the above slight account which he has 

 given me of his " new calculus," I am inclined to think it is 

 different from one that has conducted me to some singular 

 discoveries, particularly respecting the properties of functions, 



and of a"*. O", and most branches of analysis. But whether 

 our calculi be similar or dissimilai', I have thought it due to 

 Mr. Herschel to make the above quotation. With respect 

 to my own calculus, it is much more general than any I have 

 yet seen. The results of Generating Functions, which is the 

 most refined and powei'ful invention of the present age, flow 

 from it with extreme facility ; and in the cases I have yet con- 

 sidered with much superior generality. Some new views 

 having recently opened presenting a large field for discovery, 

 and the great extent of the subject, prevent me, however, from 

 now discussing it. I shall therefore confine myself to some 

 theorems relating to general differentiation and integration, 

 which I first obtained by my calculus, but which I perceive 

 may be demonstrated by common Algebra. As these algebraic 

 proofs were first suggested and are easily inferred fiom a sim- 

 ple algebraic proof of the binomial theorem, whicli, at his re- 

 quest, I undertook to discover and send to my late pupil Mervyn 

 Crawford, Esq. of Trinity College, Cambridge, I shall com- 

 mence with this demonstration, which will save me going into 

 the minutiae of the demonstrations of the other theorems. 



Algebraic Demonstratio7i of the Binomial Theorem. 



Let X + ,y be the binomial root whose first powers by the 

 ordinary rules of involution are found to be 

 X -\- y = X -v y 

 {x + y)- = J'"- + 2x V + }f 



\x + yY =. x^ -\- 3x\ij + 3.ry + y' ... 

 {x + yY ■= X* + ^x^y + Gx'y- + 4^7/' + y ^ ' 

 \x + yY = x'' + 5x*y + lOjp'y + lO^i/^ J- 5x7/ + j/* 

 (X + yf = a-' + ar^y -f 15x*f -f 20x^1/ j\- 15.ry + 6x1/'^ + y^ 

 ' Of 



