136 



E. L. De Poretit — Unsym metrical Law of Error 



Tlie constants A, B, C represent the squared q. m. errors or devia- 

 tions of the coefficients or masses I in the >ith power, from the X, Y 

 and Z axes respectively. In other words, they are the x^, y^ and z^ 

 moments of the system of coefficients I. When n becomes an infinity 

 of the second order, they are the moments of the system of vahies of 

 the limiting function ?<', and are constant and finite areas. Likewise 

 the constants Aj, Bj, etc., are the .»'^, x~y, etc. moments of the sys- 

 tem of coefficients I in the nth power, all such moments being n times 

 greater in the wth power than in the first power, as shown in my 

 article already cited {Analyst^ x, p. 97). When 7i becomes an infinity 

 of the second order, A, A^, Bj, etc., become the moments of the 

 system of values of the limiting function ?/;, and might be expressed 

 thus : 



^ =^^%^-^^""^''^^^' 



dxdydz 



fJ'J'wj' 3 dxdydz, 



(55) 



^^=^^^^.•^^•^'^''"2/^^%^^, 



and so on. 



The difterential equations (52) or (54) cannot be further simplified 

 without impairing their generality. But as they apparently cannot 

 be integrated in their complete form, we will neglect the seven ?/ and 

 6 inequalities, and thus reduce (52) to 



d^w— ^{6^-^/3 ^)dj'w —.I- 



w nfJ^d.r' 



dyW-^{S^-^ft^d;w _ -y 



w n^^dy' 



dgic—\{d^ 



A. } 



■fi^d,h(^ _ —z 

 w n§jdz 



These equations are of the same form as (IH). 

 origin to another point by putting 



(56) 



We transfer the 



Infi^dx 



y- 



infi^dy 

 6. ' 





in place of x, y, z, and assume the new constants 



2/3XdxY 



2fiXdzy 



_2j^{dyy 



"•- s^idxy '^- S.Xdyy' 



b^ = nfiXd.>-)\ fK=nP^{dyy, 

 The equations may thus be approximately reduced to 



»~ d.xdzy 



b,= n/i^idzy 



(58) 



