5T.IUTY OP A SERIES IX nNDfl^ WW&KW 215 



For example; let m =: yfey, and gi zr T ^ 7 ; and let a be Examples, 

 fust *&: tneu * VVH * become, £, c := 3-14, rf = 20*5, and 

 e — 76*7 ; and supposing x zr '3, the equation for n wil) be- 

 come fo% by the correct series, and y^ by the approxima- 

 tion : then in' order to reduce n to its true value, we may 



may make xyx' ~ [n x)' 3d x ri t very nearly, and x = ^-; 

 but y is found, in this case, •074, and x 



800 X *074 

 01014, whence x is accurately '2898 : or, x being *3, a will 



oua 



be nearly (jTJ7) X '005=: '0044, by the approximation. In the 

 second place, let a be T V • then b will be £ cm 46*875, 

 d = 1237, and e — 47200 ; and suppose x — £ ; then n bei- 

 comes *00382, ri zz '00007, and x corrected «}104. For a 

 third example, if x be \, w r e may find an approximate va- 

 lue of a by the second series, which is *0156; and this be- 

 ing necessarily somewhat too large, we may assume a zr 

 •015, whence n is found '00371, find n being '00004, a will 



be nearly — n, or '00016, and a, corrected, '01516. In the 



n 

 first and third examples, the calculations of an eminent mar 

 thematician on the continent, deduced nearly from the same 

 data, have given fpr a, instead of -0044 and '0152,, '0038 

 and '0136; while some very accurate experiments had long 

 before made the results '005 and -015. 



If we wished to investigate the properties of the returnv other eases, 

 ing branch of the curve, of which x and y are the coordi- 

 nates, it would be necessary to apply a proper correction to 

 the fluent/ x y * ; another form of a similar curve might also 

 be obtained by substituting p — y fory; and when x be- 

 comes infinite, we find, by a different mode of calculation, 

 beginning from the other end of the curve, ~ x "£ </ [p x n~ 



J} 4 j>+ v{pp— yy\ 



I am, Sir, 



Your very obedient servant, 

 \3 Feb. 1809. E.F. G. H. 



-VII. 



■ 



