110] - DIFFERENTIAL PARAMETER INVARIANT. 333 



If the given direction is perpendicular to the given plane P = 1. 

 Accordingly for q { = x, q 2 = y, q B = 2, the rectangular coordinates of 

 a point, we have P x = P y = P z = I, and for any function f(x, y, e) 



,dx) - \dy) " {-$*, 

 The projections of P on the coordinate axes are the partial parameters 



This agrees with the definition already given in 31. 



Consequently, if cos (sx), cos (sy), cos (ss) are the direction cosines 

 of a direction s, the derivative in that direction 



~ = P! cos (sx) -f -Pg cos ( s y) + -^3 cos ( S;r ) 



0F , v , 01? , v . aF / x 



= - cos s + cos $ -f cos s^, 



which is the same as equation 38 a) of 31. 



We have in this section defined the differential parameter in a 

 geometrical manner, not depending on the choice of axes of coordinates. 

 If however we take as the definition of the arithmetical value of the 

 parameter the equation 



P = 



and then transform to other coordinates x 1 , y',0', by equations 109), 

 76, we easily find by calculation that 



w, 



is equal to P, that is, the parameter is a differential invariant, as is 

 at once evident from its geometrical nature. 



If f(x, y, z) is a homogeneous function of degree n, by Euler's 

 Theorem, 



or 



nf= P{xcos(Px) + ycos(Py) + ^cos(P^)}. 



Now the parenthesis is the distance from the origin of the 

 tangent plane to the level surface at x,y,z. Calling this d, 



nf=P*, P = * f > 



