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hi the Position of a Point in Space. 127 



done in Analyst, viii, p. 4, that the origin or place of L^^ is the true 

 position of the observed point. If each L is also regarded as tlie 

 mass of a material point, and the center of gravity of these points is 

 taken as an origin, we shall evidently have a^z=.0 and a.,=iO. And 

 if the coordinate axes passing through this origin are taken to coin- 

 cide with the free axes of the system of masses L, we shall also have 

 ^=0. By reasoning similar to that followed at Analyst, viii, pp. 44 

 to 47, it will appear that (9) and (13) still hold good after this change 

 of axes, the constants a, ;^, )', etc., referring to the new axes with the 

 same meanings as before. Now in the expression for V in (13) let 

 differentials of g of the second order be neglected in comparison with 

 ar, and we have Yz=z. Also let n be written instead of "i^^ + l, which 

 is pei-missible because « is infinite. Then giving to i and j their 

 equivalents from (11), we have (13) reduced to the form 



i|.?..-i(<y-^Jr?>.-i(,4-.5>f,e-(,-(?>?X.[ = -=^^ I 



ij ^_|(<J^^^J^>-i(';,-4-^Jc?;3- (v,^5J^,^^ I =nf^^- j 



If we also write 



A =nli^{dxy, B =nf3.-,{dyY, 1 



Aj =:/«,(5^((fe)3, A^=nr;^dx{dy)^, y (15) 



\-]^=jtch.{ dy) ■•' , B 3 = ;^ // ^ (^a;) ^ dy, j 



(14) may Ite put in the form 



It will be noticed that in the expressions for A and B, ^^{dx)^ and 

 ^^{dy)~ represent the squared q. m. errors or deviations of the coeffi- 

 cients or masses L from the free X and Y axes respectively; or 

 according to the nomenclature which I adopted in Analyst, x, p. 99, 

 they are the x^ and y~ moments of the system of coefficients L about 

 those axes. The moments for the nih power are n times as great as 

 for the first power, so that the constants A and B represent the x- 

 and y- moments of the system of coefficients I in the nth. power; and 

 when n becomes an infinity of the second order, they are the a-- and 

 y2 moments of the ordinates z to the limiting surface, and are con- 

 stant and finite areas. Likewise the constants Aj and A^ are the x'^ 

 and xy'^ moments, and B, and B^ are the y^ and x'^y moments, of 

 the ordinates z. The constants in (15) might therefore be expressed 

 thus. 



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