MATHEMATICAL NOTE 13 



of the arc ^F in the other plane at A which is at right angles to 

 the plane at ABC. Let the element of area be displaced to the 

 position XYZW where AX = BY = EZ = FW = 8N. If the 

 elementary angles Z ACB and Z AC'F are denoted by a and 

 /3 then the area of the element of surface ABEF is equal to the 

 product oi AB and AF, i.e., it is Ra X R'0. If we denote this by 

 s and the area of XYZW by s + 5s we see that 



s = RR'a^, 



s + 8s = (R-\- 8N) (R' + 8N) a^. 



Therefore, neglecting products of the variations, we obtain the 

 result 



8s = (R -\- R') 8N a|3 



= s{c + c') 8N. 



But since c + c' = Ci + C2 it follows that 



8s = (ci + cz) s 8N, 

 a result used by Gibbs in obtaining equation [500]. It is used 

 again on page 280 in the lines immediately succeeding equation 

 [609] (where J'a 8Ds is replaced by y*o-(ci + C2)8NDs) and also 

 on page 316. 



If the equation of a surface in Cartesian coordinates is given 

 in the form 



2 = fix, y) 



importance for our purpose. But in order to avoid producing a wrong 

 impression the writer must point out that if a plane section is drawn con- 

 taining the normal to the surface at A, it is in general not true that the 

 normal in this plane to the curve AB at B is also the normal to the surface at 

 B. In our example where we are considering elementary arcs and areas 

 of small size, this feature may be ignored without detriment to the 

 argument. 



