234 Intelligence and Miscellaneous Articles. 



plane through any point in the surface, and at right angles to the 

 v. T plane. Its trace upon the v, T plane is 



T=0+«v, (4) 



p being indeterminate, where a is the tangent of the angle which 

 the trace makes with the v axis, or 



(IT 



(t =^- " < 5 ) 



From (3) and (5) we have 



dp=(Aja— B)dv (6) 



Calling S the slope of any element of the intersection of the 

 plane and the surface, dz being the projection of the element on 

 the v, T plane, we have 



S=^ = _J? , (7) 



dz s/dv' + dT* 



which by (5) becomes 



S=^ 1 



dv ' Vl + « 2 ' 



and bv (6) we have, further, 



8=^=2 (8) 



Vl + « a 



In determining the direction of maximum slope at any point, 

 it is evident that A and B will be constant, which gives as the 

 required condition, 



'AS _ A + Ba _ f> 



or 



A 



Substituting the values of A and B, we have 

 E 





T 



= - — =tanr (9j 



1 J 



For verv low pressures, the direction of maximum slope -1 



dz 

 becomes more and more nearly at right angles to the plane of p, v ; 

 while for high pressures this direction becomes more and more 

 nearly parallel to the plane of p, v. The direction of maximum 

 slope is constant along a line of constant pressure. 



2. To find the direction of the isentropic line at any point on 

 the surface, as related to the direction of maximum slope deter- 

 mined in (9). 



Poisson's equation, 



T«*-i= const., (10) 



is a projection of the isentropic line upon the plane of v, T, where 

 1c is the ratio of the specific heats =1*41. 



Calling «' the tangent of the angle which any element of this 



