424 HITCHCOCK. 



for a unit vector which is constant and coincides with e when /3i is 

 written for p, the right member becomes, because Sea = 0, 



In the first of these two terms, we have to distribute the operator — , 



differentiating first as if e were constant, that is, equal to a, and second 

 as if 6 alone were variable. The two results are 



k' k' ^°^^ h' ^'^%: ^"^^ 



Since Sa(f>a is the same as the second derivative of S\pFp in the con- 

 stant direction a, we have, when |3i is put for p after the differentiation 



1/7 1 d^ I d^ 



— • — Sa0a = -^ ^y- S\pFp = ^S\ -" , pFp (176) 

 Ta dh^ TadhJ '^ '^ Ta dhj^ ^ ^ ' 



which is in the most convenient form for differentiation. 



Taking next the vector — -, occurring in the second term of (175), 



dn^ 



we note that the derivative of a unit vector is always perpendicular 

 to the unit vector, hence 



J- = wp + ^v, (177) 



dh^ 



where u and « are scalar coefficients; for e, v, and p form a rectangular 

 system. Because a is a homogeneous quadratic vector, (/)p = 2o-; 

 and Sacr = 0, hence the term in xi disappears. The scalar v equals 

 -g, for 



— Sve = 

 dht 



= S \- Sv -—, bv distributing, 



dhf dhf 



= Sexe - V, by (177), 



= —g—v, because x« = <7c, and e- = —1. 



