126 h\ L. JJe torest — Uitsymrnetrical La u- of Error 



of the second order, that is, of a magnitude comparable with the 

 quotient arising from dividing a finite area by {cl.c)'^. This supposi- 

 tion extends the expansion to infinity througliout the plane XY, and 

 the small portion of the limiting surface included within the group 

 above mentioned might be regarded as approximately a plane surface. 

 {Analyst, viii, p. 42.) But for greater accuracy, we will now suppose 

 it to be a surface of the second order. Let z be restricted to mean 

 the middle one of the coefficients in this group ; we wish to find its 

 relation to ;*' and y. Let the first and second differentials of z in the 

 X and y directions be written instead of the corresponding differences 

 A in (5). Then for any coefficient whose coordinates reckoned from 

 2:=r/,j are adx and hdy, we have the expression 



z + ad^ z + bdyZ + -- d\ z + — djz + abd^d^, (12) 



and all the {2m + l)^ coefficients in the group will be successively 

 represented by assigning to a and b all the integral values between 

 — ?u and m. Suppose all the coefficients I in (9) to have their values 

 thus expressed. Collect separately the coefficients of z, d,,z, d^, etc., 

 remembering that 21u=.l. Let a^ and a.j denote the sums of the 

 products of each L into its first and second sub-indices respectively. 

 Let ^, and ^^ ^^ ^^^ sums of the products of each L into the squares 

 of its first and second sub-indices respectively. Let y be the sum of 

 the products of each L into the product of its two sub-indices. Let 

 d^ and d^ be the sums of the products of each L into the cubes of its 

 first and second sub-indices respectively. Let 7/j be the sum of the 

 products of each L into the product of the second sub-index by the 

 square of the first, and let t^^ be formed in like manner, from the 

 products of the first sub-index into the square of the second. It will 

 be found that (9) is now reducible to 



z—a^d^—a^d^z + l/i^dlz + ^fJ.,d%z + yd^dyZ=V, ^ 



— a^z+fi^d^z + yd/.—h6^dlz—^}/.2d^^—t/^d,d„z=i--- V, -^ /^gx 



— a.2Z+yd^-\- fi„d,;z—hi^d'^^—hh2d~,^—}]od^d,jZ= — - V. 



These are the two differential equations of the limiting surface, a, (?, 

 y, etc., are constant parameters. 



When the coefficients L in the given j)olynomial are regarded as 

 probabilities of error in the position of an observed point, then L^.j 

 denotes the probability that an error which occurs will fall at the 

 point x-=iaJ,i\ y=.bJy. We need not suppose, as was unnecessarily 



