482 Mr WARBURTON, ON SELF-REPEATING SERIES. 



accordance with the Diagram No. II. In this second Diagram the numerator being recurrent, 

 the respective columns equidistant from the middle column on either side of it must be the 

 exact counterparts, one of the other. Thus the high numbers 6% 7 9 , 8 9 , and 9 9 , are at once 

 excluded from the process of summation ; we have only 5 lines, that is (n + 1), instead of 9, 

 that is (2ra + l), to sum ; and the greatest Number in any line, instead of being, as in Diagram 

 I., 285,188,825, is in Diagram II. only 1,467,181 ; and the number of the multiplications to be 

 made of digit by digit, is reduced from 129 to 41. 



But let us now proceed to the improved mode of summation, depending on the employment 

 of the differences of the powers of zero, for the introduction of which into analysis the mathe- 

 matical world are mainly indebted to Dr Brinkley and Sir John Herschel, members of this 

 university. 



Then, according to the formula given by Sir John Herschel, in Jameson's Philosophical 

 Journal for January, 1820, we have, 



r a 1 i 



l 9 - 9.H + sY - &c. = 



(l + tf A'« 



v 1 + 



l + t 



9 , (23) 



\ t f f tf 

 m a'o 9 x ■ - A 2 9 - • + A 3 o 9 - A 8 9 + A 9 9 . (24) 



(i + 2 (i + o 3 (i + 4 o + o o + o 10 



and reducing these terms to the common denominator (l + t) 10 , and expanding the powers of 

 (l + t) in the numerator according to the powers of t, we obtain the terms shewn in the 

 Diagram No. III. ; which when t is equated to 1, give for the numerator the expression, 



256A'0 9 - 128A 2 9 + 64A 3 9 - 32AV + l6A 5 9 - 8A 6 9 + 4A 7 9 - 2A 8 9 + A 9 9 ; 



with a denominator 1024. . . (25) 



But on the assumption that we know beforehand that the numerator is recurrent, then the 

 columns equidistant from the middle one, on either side of it, become the counterparts of each 

 other; and the numerator acquires the form exhibited in Diagram IV. 



But then, this farther question arises. When t is equated to 1, can a simple rule be given 

 for the horizontal summation of the numerical coefficients which stand on the several lines, 

 those on the same line having all the same sign. The answer to this question is very 

 satisfactory. 



I proceed to shew that, in a Table constructed on the principle of Diagram IV, the half 

 of the sum of the coefficients in any line, less the half of the coefficient of the middle term in 

 that line, is equal to the sum of the coefficients in the following line. 



For in the developement of any two consecutive powers of (1 + t) ; say (1 + t) v and (1 + t) p ~\ 

 it is a known property, that the sum of all the terms in (l + t) p from the coefficient of t° to 

 that of tP~ v , inclusive of the two extremes, is twice the sum of all the terms in (l + ty~ l , 

 from the coefficient of t" to that of t?~ v , inclusive of the two extremes. That is to say, 

 adopting the factorial notation, 



«|-1 w + l|-l p-v\-l v\-l « + l|-l p-(u + l)|-l p-v\-l 



p_ p p . LfCfi- 1 ) .(p- 1 ) . , ( p- 1 ) . (p- 1 ) 



f (P - 1) , (P ~ . iP- 1) . (P- 1 ) I 



*| !«|1 + ]» + l|l + "• + lP -(v + l)\l + lP -»|l J- 



