VOL. XXV.] PHILOSOPHICAL TRANSACTIONS. 343 



From the nature of this series it is manifest, that if 7i be taken any uneven 



integer number, either affirmative or negative, then the series will stop or be 



finite, and the equation become one of those above-mentioned; the root of 

 which is, 



(1) yzrz^X^^^l+aa + a — 



V '^1 + aa ± a 



or (2) y = -^^^^1 + aa + a ^ -^^ ^ 1 + aa - a > 



or (3) y = „ -- —^== =z — 4v VTToo - a » 

 V V 1 4. oa — a 



or (4) y = 



T 



V V' 1 -j- aa— a V V i + aq + a 



For example, let it be required to find the root of this equation of the 5th 

 power, 5z/ + 20j/^ + 1%^ = 4 ; in which case it will be n = 5, and a = 4 

 Then the root, according to the first form, will be w = 4^ v'17 +4 1 ■ - ■ 



which is very expeditiously reduced into common numbers in this manner. 

 First i/ 17 + 4 = 8.1231, its log. is 0.9097164, the 5th part of which is 

 0.1 8 19433, the natural number corresponding to which is 1.5203 = ^ V 17 +~^- 

 Also the arithmetical complement of 0.1819433 is 9.8I8O567, to which cor- 

 responds the number 0.65 7 7 = -r — Therefore the half difference of 



-S/V17 + 4 

 these numbers is 0.4313 =: 1/. 



It may be here observed that, instead of the general root, it may be sufficient 

 to take ^ = 4.^ 2a — ;p^, whenever the number n is very large in respect of 

 unity. As, if the equation were 52/ -j- 202/* -f- l6y^ =z 682 ; then the log. of 

 la = 3.1348143, its 5th part is O.6269628, to which corresponds the number 

 4.236: also the arithmetical complement is 9.3730372, the natural number of 

 which is 0.236 : then the half difference of these two is 2 = y. 



Again, if in the foregoing equation the signs be made alternately affirmative 

 and negative, or, which is the same, if the series be of the following kind, 



, l-nra 3 , 1—nn ^^ 9—nn . , 1 — nn 9 — nn 25-n» 7 , c -l 



nv + - — - ntf + - — - X :: — 7 wv + 7, — 5 X ;; — r X ^ = w + &c = ^, its 



•^'2x3>^ '2x3 4x5-^ '2x3 4x5 6x7^ 



root will be * 



(1) ^ = 4-;^«+ V^^3I^ ^a+V-^^ITi' 



or (2) y = ^Va+Vaa-l-\.^Va-Vaa-iy 



or (3) y= ■;r^=L^= +4-V'«- -v/^^Zl* 



or (4; 5/ = -, _.,_ . + 



v' « — Vaa— 1 4/^+ '^m—. 



