578 PHILOSOPHICAL TKANSACTIONS. [aNNO 1795. 



a - 3r • 



3 "7 



^ '^ ' , the term of the multiplicand directly over it, into x , the first term in 

 the multiplier. The 2d term in the perpendicular line of products is obtained 

 by multiplying that term of the multiplicand in the next perpendicular line to- 

 wards the left, by the 2d term of the multiplier. Thus - X - X ^^ 



n — r o »• • . 1 



z^x ' , is the product of - X -^ — z^ x into - 7X . And in general, 

 if p be put for a number denoting the place of a term in the perpendicular line 

 of products, and if the terms in the multiplicand be supposed to be numbered, 

 beginning with that directly above the perpendicular line of products under con- 

 sideration, and reckoning towards the left hand ; and if the terms in the line of 



I 



the multiplier be numbered, beginning with x^, and reckoning towards the right, 

 then the product whose place is p will arise from the multiplication of that term 

 in the multiplicand whose place is denoted by p into that term in the 

 multiplier whose place is also denoted by />. The observations in this and the 

 last article are evidently general ; being applicable to any extent to which the se- 

 ries in the multiplicand and multiplier may be carried. 



l6. The laws of arrangement being thus established by the exponents, the 

 summation of the coefficients, in any perpendicular line of products, is next to 

 be attended to. And in order to do this, with as little embarrassment as possible, 

 Mr. R. puts A = n, B = n X {n — r), c = n X {n — r) x {n — 2r), D = « 

 X {n— r) X {n — 2r) X (m — 3r), &c. also a= \, b = 1 X (l — r), c = 1 

 X (I — r) X (1 — 2r), d= I X (l — r) X (l — 2/) X (l — 3r), &c. and 

 «=l,(3=l X2, y=l X2X3, ^=1 X2X3X4, &c. and then the 

 multiplicand, multiplier, and products will stand in the following manner, the 

 powers of x and z being omitted. 



a OA f^ ah f^ „ 



fir* fixr 



c 



+ , &C. 



Now the object in view, with respect to the coefficients, is to prove that the 

 perpendicular lines of products will be, beginning at 1 and reckoning towards the 



