220 PROFESSOR THOMSON'S INVESTIGATION OF A NEW SERIES FOR 



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

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



lu — udv du u dv 



dL . - „ - - 



ir V V V 



(u\ vdi 

 v) ~ 



which, by integration and transposition, gives 



fdu^u rudv 



J V V J V V 



By taking in this u = x, and xi = \-\- o^, the expression found above becomes 



X r X Ixdx _i X r^x'^dx 



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

 (8), by taking du — 2a^dx, and ^ = (1 + xy, and is found to be 



3*(1 +a;2)2"'"X/(l+x2) 





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

 tinuation being manifest, we obtain 



^ _i X 2 x^ 2.4 x^ 2.4.6 x'' , , ,q. 



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. 



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

 k to denote the fraction —^—, we get the following expression, which answers 

 best for the purposes of computation : — 



'"""i=?+?+i*^+^^+l*'^+«''^ w 



We have thus obtained the means of computing a circular arc in terms of 

 its tangent. The well known series, 



tan-i x = x — \x^ +-x^ — ^ a;? + &c (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. 



