PEIRCK. — SPACE DIFFERENTIATION OP THE SECOND ORDER. 379 



It is evideut from the defiuition just given that 



^ ' c'x- ^ dx • dy dy 



■^y-^9.-^"'-^'9y)'9^^y-^-9^^'''^-9y-)W' ^'^ 



and that Z), D,^ 12 is quite different in general from Z), D^ f2 : the order 

 of the two differentiations is material. 



If the u curves happen to be a family of parallel straight lines and the 

 V curves another family of parallel straight lines, 



D,^D,Q = hh'g^ + {hm, + l,m,) • ^^^ + m, -m,-^, (5) 



and the coefficients in this expression are constants. 



If the u curves and the v curves are identical and are a family of 

 straight parallel lines, we have 



the familiar form of the second derivative of U along the fixed direction 

 Si, which often appears in work involving the transformation of Cartesian 

 coordinates. Simple special cases of this formula are obtained by putting 

 / equal to 1,0, and ?«. 



Since li^ + '«/ = 1 5 



9h 9 nil 9mi 9li 

 9x 9y 9x 9y ^ 



PO 

 and if at any point Si and S2 ^re such as to make the coefficient of ^r— 



Id. 



in (4) vanish, the coefficient of 7;— will vanish also. Such points as 



9y 



this lie, in general, on a definite curve, the equation of which is to be 

 found by equating one of these coefficients to zero. If 5^ is a fixed di- 

 rection so that li and rui are constants, (4) takes the form (5), but the 

 coefficients are not constants unless S2 also is fixed. 



If the two variable directions s^, s., coincide, (4) becomes the second 

 derivative of the function CI taken with respect to the direction Si ; 

 that is. 



