V. Ox AX UXSYMMETRICAL LaW OF ErEOR IX THE POSITION OP 



A PoiXT IX Space. By E. L. De Forest, Watertown, Coxx. 



^HE law for space of one dimension has been treated of in my 

 articles " On an Unsymmetrical Probability Curve," which appeared 

 in The Analyst (Des Moines, Iowa), vols, ix, p. ]35, and x, p. 67. 

 The curve was obtained as a limiting form of the series of coefficients 

 in the expansion of a polynomial to a high power, special means 

 being employed to secure close approximation. Its equation is 



-TT- 1 /i I "^^j^'^—i —ax ^ 



11 ( ^ ' 



When «=Gc , this curve becomes identical with the common or sym- 

 metrical probability curve, 



xr 1 -a;'-f-26 ,^, 



^=v("2^*r ■ <'> 



I had previously shown in the same journal, vols, vi, p. 140, viii, 

 p. 3, and ix, p. 33, that the symmetrical law of error in the position 

 of a point in space of one, two or three dimensions can be obtained 

 as a first approximation to tlie limiting form of the system of coeffi- 

 cients in the expansion of a polynomial of one, two or three variables. 

 In like manner the unsymmetrical law in space can be found by 

 extension of the method so as to secure a closer approximation to the 

 true form of the system of coefficients. 



We will first consider space of two dimensions only. The known 

 formula for symmetrical differences, where u is a function of an 

 abscissa a //cc, is 



«„ = ?/+- J, +— J + _?^ ^^oH '^ ~^Z/4+etc. (3) 



"1 1 1.2 ^ 1.2.3 ^ 1.2.3.4 ^ ' 



Starting from the middle term u^^ in the series 



'^—2' ^—1? ^^0' *'l' ^2 5 



the common interval Jx between consecutive terms being the unit of 

 abscissas, this formula gives any required term z<„, where a may be 

 either a positive or a negative number. The differences A^, z/g, etc., 

 of the function u are formed from terms similarly situated on either 



Trans. Conn. Aoad., Yol. YI. 17 March, 1884. 



