398 HITCHCOCK. 



but this is equal to (125) (235) (351) (5ap) by the identity 

 (35p) (al5) - (3a5) (51p) = (351) (5ap). 



If we therefore agree to write, in keeping with the notation abeady 

 used, 



Ci235(5a, p) = (12p) (35p) (235) (al5) - (125) (3a5) (23p) (51p), (83) 

 the writing of via + wjSs for ^i gives 



(P4P5PP) = m\PaP,Pp) + mn {l2^YCm,{^a, p) 

 and, similarly, 



(P4P5P7) = viKPaP,P,) + mn il23yCi23,{,a, 7). 



The factors (457) and (456) become m (a57) and m (a56). Hence 

 the coefficient of /Se in (31) becomes 



(a57) [m{PaP,Pp) + n(123)^Ci235(5a, p)] 

 (a56) [m(PaP5P7) + n{12SyC i2Z5{,a, p)] 



which, if 7?? approaches zero, approaches the limit 



(a57)Ci235(5a, p) 



(a56)Ci235(5a, 7) 



(84) 



Expressions like the right of (83), while of geometrical significance, 

 are sometimes less convenient than determinants, (or scalar products), 

 like (PaP^Pj). We might have kept the latter form of work by 

 writing at the start, (by (23)), 



P{via + 71/35) = ni^Pa + n^P& + mnPa^, (85) 



where Pa^ has been written for 



Pa, = i [(31a) (125) + (315) (12a)] + j [(12a) (235) + (125) (23a)] 



+ k [(23a) (315) + (235) (31a)] (86) 



If we attach a similar meaning to any other P with double subscript 

 (i. e,, the result of polarizing P(fi) with respect to two vectors), the 

 coefficient of /Se in (31) approaches, by the same reasoning as before, 

 the limit 



