ESSAYS 



ON 



POWERS AND THEIR DIFFERENCES. 



BY FRANCIS BURKE, Esa. 



OF OWER, m THE COUNTY OF GALWAY ; BARRISTER AT LAW, AND A. B. Of 

 TRINITY COLLEGE, DUBLIN, 



Communimlid Sy RICHARD KIRWAN, Esq. P.R. I. A. ani F. R.S. Kc. 



Bead, November 6, IfiOB-, 



THE FIRST ALGEBRAICAL ESSAY. 



THE BINOMIAL THEOREM, 



INTRODUCTION. 



By the Binomial Theorem, we obtain a simple and general 

 formula, which represents either the result of a constant mul- 

 tiplication of a Binomial quantity by itself, or some Root of 

 such product, or the reciprocal of either of thfese. All the cases 

 are concisely expressed by the exponential notation. In the first 

 case the exponent or Index is an affirmative whole number, 

 in the second case the exponent is an affirmative Fraction, 

 and in the last case, in which the Theorem represents a Quo- 

 tient, the exponent is either a negative Fraction, or a negative 

 Integer. In the first of these, the series is often called a 

 power as opposed to a Root, altiiough " a power" is a general 

 VOL. XI. T term 



