from the development of a Binomial, 273 



which theorem is established by Abel in an elaborate memoir 

 at page 82 of his second volume. 



If we put w—h for x, and make a=A, and /3= — /*, we shall 

 have 



/ x i ,-x . d<plx—2h)2h d*a(x-3h)(3h)* , . 



a theorem given by Murphy in the Philosophical Transactions 

 for 1837> and which is usually referred to as "Murphy's 

 theorem." It appears, however, from the above method of 

 obtaining it, that it is a particular case of the general theorem 

 of Abel. But as Abel's posthumous papers — among which 

 this theorem was found — were not published till 1839, ten 

 years after the author's death, Murphy's claim is quite inde- 

 pendent, although not, as hitherto supposed, exclusive. 



The preceding method of establishing this general theorem 

 of Abel, by assuming a series of a particular form, and then, 

 as in the investigation of Maclaurin's theorem, putting, in the 

 successive differential coefficients, those values for x which 

 cause all the terms containing this variable to vanish, is very 

 short and simple ; but without the previous proof of the par- 

 ticular case of the binomial theorem, we should have had no- 

 thing to suggest to us the peculiar form selected. This sug- 

 gestion, however, once supplied, we may, instead of examining 

 each differential coefficient in succession, as above, and thence 

 inferring the general uniformity of the law of the coefficients 

 A, B, C, &c. by induction, — instead of this, we may proceed 

 at once to demonstrate the general property that 



d n w{x—m ^) m - 1 



~~dti» ( A v 



is zero for #=w/3, provided n < m ; and that it is 1 .2.3....W, 

 for n=m; which property will establish the above values of 

 A, B, C, &c. rigorously and universally. 



To prove it, conceive x in (A.) to be changed into x + h, 

 the binomial to be developed, and the series multiplied by 

 x + h: the coefficient of h\ in the result, will, of course, when 

 multiplied by 1 .2.3...W, be the nvh differential coefficient (A.). 

 It is easy to see that this coefficient, that is the coefficient of 

 h n 9 is 



(m — l)(m— 2)(w — 3)....(w— ») ; 



i.g.g...., — '-i—w 



\m— n— I 



(m-l){m-2)....(m-n +l) 



+ 1.2....(»-1) (*-»0)" 



* The more general form of the theorem is got by putting 

 x=x—», »=nh, /3=- h. 



