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 — , 



dhf 



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



as if 6 alone were variable. The two results are 



Jl I . SacAa+^.Sa^l^. (175) 



la aha -'o' tt"t 



Since Sacpa is the same as the second derivative of SXpFp in the con- 

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



which is in the most convenient form for differentiation. 

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



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

 to the unit vector, hence 



~ = vp + vv, (177) 



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

 system. Because cr is a homogeneous quadratic vector, 4>p = 2a; 

 and Saa = 0, hence the term in u disappears. The scalar v equals 

 -g, for 



-^ Sve = 



dhe • ■ 



,^ edv ^ ^ de . ,. ., 

 = o — — r ^^ ~rr) by distnbutmg, 

 dhf dhf 



= Sexe - V, by (177), 



= —(J —V, because x^ = 9^, and e^ = —1. 



