in the Position of a Point in Space. 125 



In this way the symmetry with respect to u^^^ is preserved, ami J.,;, 

 is supposed to be located at ^'q.q. 



Denoting by L tlie coefficients in a polynomial of two variables 

 /• and s, we locate these coefficients Avith reference to any assumed 

 rectangular axes by coordinates proportional to the exponents of the 

 variables, and give the polynomial a square form by adding terras 

 with zero coefficients if necessary, so that the whole number of terms 

 in it is (2?/« + l)". Denoting by I the coefficients in its expansion to 

 the nth power, we locate them in like manner with reference to the 

 same axes. The polynomial is supposed to have been originally 

 divided through by the sum of its coefficients, so that ^X=l and 

 ^7=1. The first power and its expansion may be written 



X"!-,. XZ"-,n (La., r S"), ^Ill'L XZ-ran (4.6 ^" s') . (S) 



Then as shown in Analyst, viii, pp. 9 and 41, with only a change of 

 notation, the coefficients I in any square group of (2m + 1)^ terms in 

 the expansion will be connected with the coefficients L of the given 

 polynomial by the relation 



I 



XZ1,.,XZ':. (*L_„,_, W.),o-W =;^^^- J 



The common unit intervals between consecutive coefficients in the 

 polynomial or its expansion are Jx and ziy, and a and b are integers 

 which, used as sub-indices, locate any L at the distance aJx and 6z/y 

 from the axes of reference, while i and ,;' are similar integers used to 

 fix the position of the middle coefficient of the supposed group of 

 (2wi(-f 1)- terms in the expansion, with regard to the same axes. 

 This middle coefficient then is l;^j, and its coordinates are 



x=iJx, y=J^y. (10) 



When the exponent n is made very large, or infinite, the coeffi- 

 cients I become ordinates z to the surface which represents the lira, 

 iting form of the expansion, and we suppose them to be set closer 

 together so as to be consecutive. This Z/cc and Jy are reduced to 

 dx and dy, and (10) becomes 



x=idx, y=Jdy. (11) 



The extent of the group of (2m + l)^ coefficients / under considera- 

 tion is infinitesimal in comparison with that of the whole {2mn + l)^ 

 terms in the expansion, especially as we shall i-egard n as an infinity 



