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 already 

 used, 



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

 the writing of ma + n/35 for ^i gives 



(P4P5PP) = m\PaP,Pp) + mn (123)2^235(50, p) 

 and, similarly, 



(P4P5P7) = m\PaP,P^) + vin {123)^0 uz.^a, 7). 



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

 the coefficient of jSe in (31) becomes 



(a57) [miPaP,Pp) + n(123)2(7i235(5a, p)] 

 (a56) [miPaP^P^) + n{123yCi2Zf>{,a, p)] 



which, if m 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 {PaPiP7). We might have kept the latter form of work by 

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



P(ma + 71/35) = m^Po. + n^P^ + mnPa^, (85) 



where Pa^ has been written for 



Pas = i [(31a) (125) + (315) (12a)] + j [(12a) (235) + (125) (23a)I 



+ 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{p) with respect to two vectors), the 

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

 the limit 



