of the Orbits of Double Stars. 281 



convergency of the series may be most easily investigated by 

 introducing instead of A, B, C, &c. the sums of the reciprocal 

 even powers of numbers ; if 



1 1,1 



a = 1 +-^ + -gT + ^r 



1 1 1 

 ^= 1 + "27 + "3? + -V* 



_ 1_ J_ J_ 



it is well known that 



2A 3 



a = "t 



1 2 



O = TT 



1 234. 

 2=C 6 



C = "IT 



12 3 456 

 consequently, 



2a X ^h x^ 6 c X* M x"^ 



In the distant terms of the series the ratio of two successive 

 powers of — , say ^ — J , and \ — j , whose 



coefficients let be represented thus : 



2a J 2/* + 2 

 — —VI and — «, 



It TV 



will more and more approach to unity, so that the series will 

 always: converge for x< tt. Making x = ^ tt we have x ( i "■) 

 — ^ IT, consequently 



1 3 _ 2b 3c 4(f 5e 

 Y'^-o+-^ + ~^+ 26 + -^ 



Within the limits x = and x = i tt, the change of this 

 latter function is only the fourth part of that of the quantity 

 2x — sin 2 X. This change may, however, be still more di- 

 minished, and the transcendental function to be introduced 

 may be made still more like a constant quantity by combining 

 it with a divisor by which the first power is made to disap- 



o -r* .^ Sin 2 iZ" 

 pear. If we choose sin .r for this purpose, we have . — -j — 



exactly the same quantity employed by Gauss in his Theoria 



Motux, for solving the problem of determining the elements 



N.S. Vol. 10. No. 58. Oct. 1831. 20 from 



