B I N 



BELLOWS Hydrostatical.See Fluids pressure of. 



BINOMIAL THEOREM. 



This series, in its most simple form, is as follows : 



(a 4- b) n = a n 4- na " ' b + . !Ln2 a w - 8 o 2 -f . 



a -* o 3 -f , M - 1 . n 2 . n .. 3 a-*64 4. & c . where n is a 

 ^ 3 4 



whole number or fraction, positive or negative. 

 If b be negative, the odd powers of b will be also negative. 



f j. n 1 n 2 n n 2 



Cor. 1. The "* term of the series is n. - . ^ .... -- - . 



Cor. 2. If n be a positive whole munber, the series will consist of n -J- 1 

 terms, but in every other case, the number of terms will be unlimited. 



Cor. 3. If n be $ whole positive number, the whole sum of the indices = 

 .ww-f 1. 



Cor. 4. If n be a whole positive inuuber, and b also positive, the sum of 

 the coefficients of (a + b) n = 2 n j but if b be negative, the sum of the 

 coefficients ~ o j this appears by expanding the series, and maiding a 

 b. 



Cor. 5. If we call the index , and put = O,, and let A, B, C, D, 



&c. represent the 1st, 2d, 3d, &c. terms of the series, with their proper 

 m 



signs, we shall have (a -f- o) 



This is the most convenient practical form in the case of fractional or 

 negative indices. 



= 4- -- + llk- &c - 



