PEIRCE. — SPACE DIFFERENTIATION OF THE SECOND ORDER. 383 



and we may write in this case 



D,D,n = hlp. 



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

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

 of level.* 



When s-i and s^ are perpendicular to each other, we have in general 



9'^Q, 9'^n „ 9'^Q, 



2 3x^ 9x • Sy 9y 



(16) 



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



jj,^ u + n,^ u - ^^, + ^^, + ^^^ ^ ^^ + j^ ^^ ^^ 



_ 5^ 9^_\ ^9j>ni ,9n_\_9li 9^ , 

 9x^ 9y'^ li 9y 9x mi 9x 9y ' 



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

 wholly independent of the {^articular system of rectangular coordinates 

 which may be used. 



Spack Differentiation. 



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

 (si, S2) be defined by the direction cosines (4, wij, n,), (4, m.2, n^), where 

 li, nil, ^i> hi ^2> ^2 3,re any six single-valued point functions which 

 satisfy the identities 



li^^mi^ + n^=\, l.'' + m^^^n.^=\, (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 and second orders with 



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



