PEIRCE. — SPACE DIFFERENTIATION OF THE SECOND ORDER. 383 

 and we may write in this case 



X), t D,n = k/p. 



This expression gives the rate at which the maximum slope of the surface 

 the coordinates of which are (x, y, O), changes as one goes along a line 

 of level.* 



When s, and s 2 are perpendicular to each other, we have in general 



o 5 2 o „ t 9 2 n 79 9 2 n 



3 dx 1 dx ' dy dy 



( 9m, 9» h \9n ( 91, 9l,\9Q 



and since I, 2 + m, 2 = 1, 



91, 9m, 9h 9m, 



dx dx dy dy 



So that if we add together (7) and (16) we shall get 



s i * 2 9x 2 9y 2 m, 9y 9x 1, 9x 9y 



_9\i 9 2 n l \ 9m x m 9a \_9h9n 

 9x z 9y 2 I, 9y 9x m, 9x 9y 



It is evident that the values of the space derivatives defined above are 

 wholly independent of the particular system of rectangular coordinates 

 which may be used. 



Space Differentiation. 



At every point of the space domain, R, let two independent directions 

 (s„ s 2 ) be defined by the direction cosines (l„ m„ n,), (l 2 , m 2 , « 2 ), where 

 l„ m„ n,, l 2 , m 2 , n 2 are any six single- valued point functions which 

 satisfy the identities 



I, 2 + m, 2 + nf = 1, Z 2 2 + m 2 2 + nJ= 1, (17) 



and have finite derivatives of the first order with respect to the coordi- 

 nates x, y, z. If, then, Q is any single-valued function of the coordinates 

 which within R has finite derivatives of the first aud second orders with 



* Boussinesq, Cours d' Analyse Infinitesimale, T. 1, f. 2, p. 236. 



