Vector Differentials. 197 



3. If two vectors at right angles to each other and to the 

 normal be operated on with %x~ -m 2 they will still be at 

 right angles. 



4. If* x e= 9 6 an d X^~9'^' ^ nen <7=— Sf/Ve and 



5. With the notation of Art. 4, it may be proved that 



-St 2 V . <jyri= -StjV . (£t 2 + 00 1 t 2 --<£0 2 Ti. 



6. Putting a for S/u,%/a and 6 for SX^/Lt, while <r^\=<£ A 6fy>> 

 we may establish these six results : 



(a) <£\v= —X f X^ = ^( m i~ m 2 2 )~~ m 2 v Xt Jb ~~' c v(ai-m2); 



(V) ^(2^-m 2 )A = — ^A ■+■ \/m 2 (a + m 2 ) — XJm v ; 



W YV%X = ^X-^(« f m +br T ^^c x v X ^-c^- 



1 f ) — r-^ z=[ae + zu , — a -77 \c — m^m l — (a + ?»., ) r : 

 KJ ' cin \ dm dl 



from which may be deduced the eighteen constituents of 

 d^K and d^p. 



7. If v, v ly and v 2 are unit-normals to three orthogonal 

 families of surfaces, so that yy x =.gv\ and ^j/ 2 = </i/ 2 , with similar 

 expressions for g u g x ' and g 2 > 92 -> dv may be expressed in 

 terms of the three normals and the six g's (see Ex. 4). 



8. If Sp(<£ + P)~ r p= — 1, where cj> is self-conjugate with 

 constant constituents, UVP satisfies (21). Thence may be 

 found the distribution of electricity on an ellipsoid by means 

 of (22) . (In differentiating we may treat <fi + P like a scalar, 

 that is 



d[(<t> + T>)-ip]=(<t>+P)-\ip-(4> + V)-wv). 



9. Of <f)d(j and <j)'dp only one can be integrable. 



