130 /i". Z. De lorest — Unsymmetrical Laio of Error 



cl.i'dy 



2;rK^K,^(*,62) 

 and putting zzziLd^dy^ we have 



('^-^6;)"'"-'('^7ir-'^-"'^-"*'.(-*) 



Z=i 



This final equation of the surface sought is the product of two 

 functions like Y in (1), one in a; and the other in y. The intersection 

 of the surface by any vertical plane parallel to either the X or the Y 

 axis will be a curve whose ordinates have the form (1) multiplied by 

 a constant. Differentiating (84) we get 



dT. „(a^h-\ \ dZ /a:b-\ \ , , 



Tliese become zero for 



■"=-«,' "^-h <'"* 



at which point Z is a maximum. They are also zei'o when Z=0, and 

 this, as we know from the properties of the curve (1), will occur when 

 cc=: — rtj6, or when !/= — «2^2' or when a;=:±co,the + or — sign 

 being taken according as «, is + or — ; or when yr=±x> according 

 as «o is + or — . The intersection of the surface by any horizontal 

 plane is a closed curve of contour surrounding the vertex-point (-'56). 

 Denoting by Z' the height of this plane above the XY plane, and 

 writing 



e=27rZ'K,K.^(6j/^2), (37) 



the equation of the curve of contour, or of its projection on the XY 

 plane, is 



Neither the x nor the y can in general be explicitly expressed, one 

 as a function of the other. But if «, or «2 i^ infinite, the surface 

 becomes symmetrical in the .'■ or y direction respectively, and the 

 form of the curve of contour is simplified. For instance, with '^2=: x> , 

 we have the identity 



{Analyst, ix, p. 165) ; and (38) may be reduced to 



y^=2bA -T-! log ( 1 + — i- )-«,.'•—, *- V. (40) 



og e * \ a J) J ' log e i ^ ' 



