PEIRCE. — DERIVATIVE OF A SCALAR POINT FUNCTION. 345 



and 



d 2 u ,„ d 2 u , a d 2 u , „ d 2 u , . d 2 u , „ d 2 u , . d 2 u 



ds 2 dx 2 dy z dz 2 dx-dy dy-dz dx - dz 



duf , dl dl dl\ duf , dm dm dm\ 



dx\ dx dy dzj dy\ dx ' dy dz ) 



, duf 7 dn , 3^ , dn\ ,„_ 



If / is a direction defined by the cosines I', m\ n\ 

 d 2 u „, d 2 u , , d 2 u , , 3 2 m 



~~ 11 ■ T—5 + 7WW2 • T-S +»W 



a/ -as a^ a/ a* 2 



dx-dy dy-dz dz-dx 



,dufy dl , dl , dl\ duf ,, dm , dm , dm\ 

 dx\ dx dy dz) dy\ dx dy dz J 



duf lf dn . dn , dn\ , „. 



and it is clear that the order of differentiation is usually not com- 

 mutative. Derivatives of this kind are often found in differential 

 equations of orders higher than the first which define functions in 

 terms of simple curvilinear coordinates. 



If for instance spherical coordinates are to be used, the second 

 derivative of u taken in the direction in which 6 increases fastest is 



d* U 2Q 2 J 1 ^U 2/3-2 . ^U . 2n 2d' 2 U 2A • . 



—^ • cos 2 cos J <£ + t-t 2 • cos^fl snr y + —z ■ sm^+ - — — • cos'' 6 sin <A cos <b 

 dx 2 dy 2, dz 2 dx-dy 



2d 2 u ■ n a , 2 d' 2 u . a . du . 



— - — — • sin v cos cos <b — - — — • sin 6 cos v sin <f> — • sin v cos 6 



dx-dz dy-dz r- dx 



sin sin $ r--cos0 (14) 



r - dy r ■ dz 



and this, which contains derivatives of the first order, is in sharp con- 

 trast to the second derivative of u taken in the direction r, which is, 



— -o • sin 2 6 cos 2 </> + t-s • sin 2 ^ sin J <£ + -r-5 • cos 2 + - — — • sin 2 sin 6 cos<£ 

 dx dy* dz dx-dy 



+ - — — • sin cos 6 sm<b + z — =- • sin cos 9 cos <i. (15) 



dy-dz dz-dx ■ 



