VOL. LXXII.] PHILOSOPHICAL TRANSACTIONS. 317 



1 6 2+ 32 



This, by resolving two terms into one, becomes | + . — 4- 



1.2.3 ' 3.4.5 ' 5.6.7 

 — &c; and as the terms of the given series continually approach to unity, the 



1 16* 24 32 1 



correction, to be added, is -, consequently t g g - 4- ^-j- - + — — — &c. + - 



is equal to the sum of the given series; but by prop. 1 part 1, the sum of the 



series -—-3 + 3-^5 + jfg^} + &c> is ec l , ' ial to 8s "~ ? ( s bein g tne hv P- lo g- 

 2) consequently the sum of the given series is 8s — 14-. 

 Exam. 2. Required to find the sum of the infinite series 



hi _ hi + 3 _^i _ til + &c. 



1.3 3.5^5.7 7.9 ' 



4 8 12 



This series, by resolving two terms intoone, becomes — — - 4- ■ 4- 



1.3.5 ' 5.7.9 ' 9.11. 13 



4- &c. and as the terms of the given series continually approach to -, the cor- 



1 48 12 1 



rection, to be added, will be -, therefore — — - 4- y^~~a + q u - ^ + &c - + g 



is = to the sum of the given series; but by prop. 1 part I, the sum of 



4- — U : — J- &c. is equal to - s + - (s being a circular arc of 



1.3.5 '5. 7. 9~ 9- ll. W T l * 8 



45°, whose radius is unity) hence the sum of the given series is - s + -7- 



This method is not only applicable to those cases, where the given series re- 

 solves itself into another, whose sum is either accurately known or can be ex- 

 pressed by circular arcs and logarithms, but also to those cases where we want 

 to approximate to the value of the given series, as it must, in general, be neces- 

 sary first to render the terms of the series converging, by collecting two into 

 one, before the operation of approximation begins, and consequently a correction 

 of this latter is necessary in order to exhibit the value of the given series. 



XXF1. A new Method of finding the Equal Roots of an Equation by Division. 

 By the Rev. John Hellins, Curate of Constantine, in Cornwall, p. 41 7. 

 Theorem 1. — If the cubic equation x 3 — px 2 4- qx — r = has two equal 

 roots, each of them will be x = .rr^j^r- 



Demonst. For, call the 3 roots a, a, and b; then, by the composition of equa- 

 tions we shall have * 3 ~ 2 £}x* + '^x — aab = O, where la + b = p, aa 4- 

 2 a / } = o, and aab = r; which values being written in our theorem, we have x 

 pq-jr . a 

 '-pp — bq 



Exam. If the equation x 3 + 5x 2 — 32jt + 36 = O has two equal roots, it is 

 proposed to find them by the above theorem. 



Here p = — 5, o = — 32, and r = — 36; these values being written in 

 _ ... x -30 - 9 x _ 3b I6'0 + 324 484 . , 



the theorem, we have 2 x -o - x - 32 = 15+W = 242 = 2 > vvhich be " 



