PEIRCE. — SPACE DIFFERENTIATION OF THE SECOND ORDER. 379 

 It is evident from the definition just given that 



^ r, , , 9 2 ci „ , N 9 2 ci 9' 2 ci 



(j.Qk. % 9h\ 9ci ( dm 9m x \9ci ... 



\ cto; dy J 9x \ 3x dy J dy 



and that Z> s Z> 5 fi is quite different in general from D Sa D s Cl : 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, 



~ .. , , 5 2 n ,, , x 9' 2 ci 9 2 ci 



D h D h Q = hh'g^ + (km* + km,) ■ g^ + m, • m 2 ■ ^-, (o) 



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 



n H Q ~k g x * + z kmi 9x . dy + m i 9y 2 > W 



the familiar form of the second derivative of Cl 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 m. 

 Since I 2 + m 2 = 1 , 



9k 9 nil 9m x 9l\ 



9x 9y 9x 9y ' 



9ci 



and if at any point s x and s 2 are such as to make the coefficient of — 



oq 



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



dy 



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 s x is a fixed di- 

 rection so that I, and m, are constants, (4) takes the form (o), but the 

 coefficients are not constants unless s 2 also is fixed. 



If the two variable directions s l5 s 2 coincide, (4) becomes the second 

 derivative of the function Cl taken with respect to the direction s, ; 

 that is, 



