384 PROCEEDINGS OF THE AMERICAN ACADEMY. 



respect to these coordinates, the derivative of Ci at the point P, in the 

 direction Si, is the value at P of the quantity 



A i2 = ^1 • X- + /«! • 7^ + "l • 7=j- . (18) 



^ c/x dy dz ■' 



Through the point P passes a curve of the family defined by the 

 equations 



dx dy dz , ^ 



and this curve indicates the direction s^. If on this curve a point Q be 

 taken near P and in the sense of the direction S25 the limit, as Q ap^ 

 proaches P, of the quantity 



PQ 



may be represented by [2?^ Z), fi]^ and this is the second directional 

 derivative at P of O taken with respect to the directions Si and s^ in the 

 order given. It is evident that 



7^ r. r^ 7 7 ^'n 5'n 5^0 



dx-dy dydz ^ dz-dx 



9h , 5/1 



dx dy 



+ 4 • >r + '"2 • 7^- + w 



9h\ 9^ 

 o'2; y dx 



, , 5m, c'/tti 9mi \ 90, 



+ 4 • ^=; \- >«2 ■ ^^j 1- «2 • 



c).r 9y ' 9z ) 9y 



and that this is not equal to D^ Dg D. 



If the directions s^, s^ are fixed, the six direction cosines are constants, 

 the last three terms of (21) disappear, and the coeflTicients of the other 

 six terms are constant. If the fixed directions Si, So coincide, (21) re- 

 duces to the familiar form 



i 



