220 PROFESSOR THOMSON'S INVESTIGATION OF A NEW SERIES FOR 



The integral of the second member of this, in the form that will suit our pur- 

 pose, will be obtained in perhaps the easiest manner by means of the formula, 



, /ti\ _vdu — udv _du u dv 



\V/ V^ V V V ' 



which, by integration and transposition, gives 



fdu^u rud_v 



By taking in this u = x, and » = 1 + ar^, the expression found above becomes 



, X r X Ixdx 1 X Cix^dx 



tan ■ a; = -—- — j + / ■ . . - — ; — = , or tan x = -r— ; — 3- + I-tt—. — srs • 



I +x^ 'J I + x' I +x" 1 + x^ \/ {\ + 3?)^ 



The integral of the last term of this is obtained in a similar manner, from formula 

 (8), by taking du = 2afdx, and ® = (1 + ^Y, and is found to be 



2 x^ ■ 2A rx*dx 

 3'(i +x^y "*" 3j{i+3^y 



It is plain that this process may be continued without limit ; and, the law of con- 

 tinuation being manifest, we obtain 



♦ -1 a; 2 a;' 2.4 x^ 2.4.6 a;' , ,., 



This is the series proposed to be investigated ; and, for giving an arc in the first 

 quadrant, it requires the addition of no constant quantity. 



When a; is a fraction -, the foregoing series may be exhibited, after some mo- 

 difications, in the convenient form, 



g ^ + f\ ^Sp^ + q'^S.byp' + qO ^ i ^ ' 



By putting A, B, C, &c. to denote the successive terms of the last series, and 

 k to denote the fraction -r?^^, we get the following expression, which answers 

 best for the pxirposes of computation : — 



^"-i = PT^ + ^A+i^B + ^AC+&c (11) 



We have thus obtained the means of computing a cu'cular arc in terms of 

 its tangent. The well known series, 



tan-' a; = a;_la;3 +^^5 _ 1 .7 ^. &(. (12) 



given, first by James Gregory, and afterwards by Leibnitz, serves the same pur- 

 pose, but is far inferior in practice. Like (12), the series above investigated, con- 

 verges the more rapidly, the smaller the tangent is in comparison of the radius. 



