16] VARIABLES AND FUNCTIONS. 25 



The operator i-r-+j-^-+k which gives the vector differential 



ox oy oz 



parameter of a function, was denoted by Hamilton by V. (read Nabla). 



If f(x t y, z) is a homogeneous function of degree n, by Euler's 

 Theorem 



/. y y 



nf= x^- + y~ + z^-. 

 dx J ty 8z ' 



or nf= P {x cos (Px) + y cos (Py) + z cos (Pz)}. 



Now the parenthesis is the distance from the origin of the 

 tangent plane to the level surface at x, y, z. Calling this 8, 



or the parameter of a homogeneous function is inversely proportional 

 to the perpendicular from the origin to the tangent plane to the level 

 surface. For example, if n= 1, 



V = ax + by + cz, 



x) = a, Pcos(Py) = b, Pcos(Pz) = c, P= 



The level surfaces are parallel planes, and the parameter is con- 

 stant, 



V is proportional to the distance of the level surface from the origin. 



Pcos(Px)=, 



P /5? 5J2 



* / or ?/ s 2 



-5 +^ + 



\/ c 2 ' ^ 2 ' 2 



For the surface, F=l, 



1 





/^ ++ ^ 



V o a a a,? a 3 2 

 a familiar result of analytic geometry. 



