212 Prof. Gauss on the Representation of the Parts 



is a given uniform function of the coordinates X, Y, Z. Let 

 us suppose that we obtain by differentiation 



dty = edx+gdy+hdz 

 rf* = EtfX + GrfY + H</Z. 

 where e, g 9 h will be functions of x, y, z, and E, G, H functions 

 ofX,Y,Z. 



The considerations by which we must reach the end here 

 proposed being, although by no means difficult, yet of rather 

 an uncommon kind, we shall endeavour to give to them the 

 greatest possible perspicuity. Between the two corresponding 

 representations on the surfaces whose equations are \[/ = 0, 

 and * = 0, we will assume six intermediate representations 

 or planes, so that eight representations will come under con- 

 sideration ; viz. Considering as corre- 



sponding the points 

 whose co-ordinates 

 are respectively = 



1. The original on the surface, the equa- 1 



tion of which is v(/ = j <r '*^' 



2. The representation in the plane x 9 y, 0. 



3. t, u, 0. 



*• pi q, o. 



5. P,Q,o. 



6. T,U,0. 



7. X,Y,0. 



8. The representation on the surface, the \ v y 7 

 equation of which is * = J ' ' * 



We will now compare these different representations merely 

 with regard to the relative position of the infinitely small linear 

 elements, without any regard to the relative magnitude, and 

 we shall consider two representations as similarly situated, if of 

 two elements proceeding from the same point, the one to the 

 right, in one representation has its corresponding element in 

 the other, likewise to the right ; in the contrary case, we shall 

 call them reversed. In the plane from No. 2 to No. 7, that 

 side on which are the positive values of the third coordinate 

 is considered as the upper one, but in the cases of the first 

 and last surfaces the distinction between upper and lower side 

 depends only on the positive and negative values of ty and * 

 as has been before explained. 



Now it is in the first place clear, that for each place of the 

 first surface where, x and y remaining the same, a positive in- 

 crement of z carries to the upper side, the representation 2 

 will be similarly situated with the representation 1 ; this will 



evidently 



