Dr. Bankine on Rational Approximations to the Circle, 421 



kept permanently flowing through the wire. It is very doubtful 

 whether in practice a smaller variation than this would be avail- 

 able for practical signalling by any of the automatic plans yet 

 invented for aerial lines. This would give for the duration of the 

 shortest signal 0*0112 second, equivalent to about 90 signals per 

 second, or about 5400 per minute ; on the Morse alphabet this 

 would correspond to 360 words per minute. On a line of 1000 

 kilometres this speed would be reduced to 90 words per minute. 

 Thus Professor Wheatstone's automatic instruments, alluded to in 

 the beginning of this paper, may be expected to transmit at their 

 full speed over a line 500 kilometres (or say 300 miles) long ; 

 but their speed would necessarily be reduced on a line of double 

 that length, unless, by augmenting the section of the conductor, 

 the value of k be reduced. The value of c would at the same 

 time be increased by doing this ; but, as will appear from the 

 formula given above for the suspended wire, its value will not be 

 materially affected within practical limits. The value of k in 

 British electrostatic measure for one foot of No. 8 iron wire is 

 not far from 7 '7 x 10 -14 , whereas the value of k in British measure 

 for a foot of the French wire is about 9 x 10 _u . This differ- 

 ence in the value of k makes a difference of about 15 per cent, 

 in the length of the line over which a given speed of signalling 

 could be maintained. 



By similar means we can calculate the number of words per 

 minute which could be transmitted by any system over any given 

 line. It is to be hoped that experiments will soon be made 

 which will determine the value of c in such a way as to supersede 

 the rough approximation arrived at in the present paper. 



LVIII. On Rational Approximations to the Circle. 

 By W. J. Macquorn Bankine, C.E., LL.D., F.R.SS.L. %E.* 



1 . npHE common method of approximating to the ratio which 

 J- the circumference of a circle bears to its diameter by 

 means of a series of inscribed and circumscribed regular polygons, 

 though practically convenient, is unsatisfactory as regards the 

 illustration of certain general principles, because of the irration- 

 ality of the successive steps of the approximation, — the cir- 

 cumferences of the whole of the regular polygons, with the 

 exception of the inscribed hexagon and circumscribed square, 

 being computed by successive extractions of the square root, and 

 expressed by means of surds. 



2. The object of the present paper is to show how, by using 

 irregular inscribed and circumscribed polygons, an indefinite 

 * Communicated by the Author. 



