384 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 direction s x , is the value at P of the quantity 



7 5o 50 9n 



D 12 = k • ^- + m, • — + Wl • _. (18) 



1 dx dy dz ' 



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

 equations 



dx dy dz ,„.. 



T = —=—> O 9 ) 



l 2 m 2 n 2 



and this curve indicates the direction s 2 . Tf on this curve a point Q be 

 taken near P and in the sense of the direction s 2 , the limit, as Q ap- 

 proaches P, of the quantity 



P<2 



(20) 



may be represented by \_D S D s W\ p and this is the second directional 

 derivative at P of Q taken with respect to the directions s x and s 2 in the 

 order given. It is evident that 



„ „ 7 7 5 2 o 9 2 n 9 2 n 



J) S2 D s n = l 1 l 2 -^- 2 +m 1 m 2 ^ + n 1 n 2 --^ 



, x 5 2 n . 5 2 o , _ 7N 5 2 o 



+ (/^a + Ui) - - + (Wi« 2 + m 2 rc,) ^— ^- + (w x / 2 + rc 2 4) ^r—^ 

 dx'dy dy 9z dz'dx 



V 9x 



+ 4 ' «- + »*s * «~ + »•: 



3d : 5'A 5ft 



cty 5z / ca; 



/ . 5m t 9m x 9m 1 \ 9CI 



\ 9x 3y dz J 9y 



( 9n x 9n x 9n l \9CL 



\ 5a; 5# dz J dz 



and that this is not equal to D s D B ^ O. 



If the directions s x , s 2 are fixed, the six direction cosines are constants, 

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

 six terms are constant. If the fixed directions s x , s 2 coincide, (21) re- 

 duces to the familiar form 



