432 Mr. Ward on Mr. J. Herapath's Demonstration. 



P.S. The turner who makes the staff" A B D G may leave 

 it capped ^t top, and make it cylindrical about half its length 

 from the top, in order that the arm may not slide down the 

 staff" when it ought to remain fixed. 



LXX. On Mr. J. Herapath's Demonstratioii of the Binomial 

 Theorem. By Mr. L. T. Ward. 



To the Editor of the Philosophical Magazine and Jotirnal. 



Sir, 



AT page 323 of the Philosophical Magazine for last month, 

 Mr. J. Herapath, in his demonstration of the Binomial 

 Theorem for integral exponents, tells us that the product of 



(g_ 1) of the quotients n, -—, -^, &c., is equal to the co- 

 efficient of the qih. term in the expansion of (.r + j/)" when q, 

 of course, is any whole number greater than 2. This of itself 

 amounts to no proof. In oi'der to complete the demonstra- 

 tion it must be shown that n x -^ x -j-, &c. to (^ — 1) fac- 

 tors, OUGHT to equal the co-efficient of the qih term in the ex- 

 pansion of {x + yfi and this may easily be done by induction 

 from the operations marked B. 

 Thus 2 X ^ 



SXg-; Sx^-Xg^. 



4 X |; 4 X I X f ; 4 X I X I X i. 



oXg'j "SXg^x^; ox-gXi^x^; oXttX^x^^Xj-. 

 &c. &c. &c. ' 



n — 1 rt— 1 n — 1 u—\ » — 2 n — 'i 



nx --;«x— X— ;«x— x-^x— ... 



(« — 1 71 — '2 1 \ 



n X — — X -^r- ••• — ) 

 2 3 n / 



It is easy to perceive how these results are obtained ; and 

 on comparing them with the co-efficients ol'the powers marked 

 A, it is evident that 2 x | = the co-efficient of the 3d term of 

 the 2d power ; that 3 X | and 3 X | x g^ = the co-efficients of 

 the 3rd and 4th terms of the 3rd power; and so on of the rest; 



, , „ , 11— I n — 1 n — 2 M — I 



and therefore, that 7i x — ^ ; n x ^— x —77- ; n x — ^ x 



'^—— X - — , &c., which are deducible from the preceding steps, 



properly denote the co-efiicients of the 3rd, 4th, 5th, &c. terms 



in the expansion 'of {x + 3/)". 



I am, sir, yours, &c., L. T. Ward. 

 Wisbech, June fi, i82.=,. ' LKXl. A 



