PHILLIPS AND MOORE. 



LINEAR DISTANCE AND ANGLE. 



55 



which is therefore equal to the former. We can write this result in 

 the form ^ 



K,i. .a„ ^,,1 . ./3J = Ml A,.. A, B, B,..B,] = ^{CB,. .B,) D (1) 



C being the product of any combination of n — g of A's and D the 

 product of the others such that 



[A, A2....A,] = [CD] 



li p-{- q < n we take the part of A which contains all the ^4's and 

 B's in one ?i-rowed minor. The result is 



A2= ^(ap+i . . .a„ ^n-p+i • ■ •/3„)[^i . . .Ap Bx . . .B^ [i^+i . . ./3„_p]. 



Hence we have 



Ui...^p -B1...5J = [a^+i.. .a„ /3g+i. . ./3J = -(ap+i. . .a„S) 7 (2) 



where S is any combination of p B's and 7 the remaining ones so 

 arranged that 



[/3,,i.../3j= [7^]. 



If instead of the determinant A we use the determinant 



(11 «i2 "hi 



«pi 

 S+i' 1 



O-in 1 

 = 611 











Ip+l- 1 



0... 



<^p+h 2 • • ='=Ctp+li n 



^Cl«l 



hi2- 



■ ^nn- ■ ■ 



^q+h 1 



Oq2 Oqn. . . 



Pq+h 2 • • • -l^q+h n 



■^n 



(3) 



(3,a. /3„2 



when 2) -{- q > n, we obtain the expression in the form 



[AiA2...A,B,B2...Bp] = ^{A,A2...Aj,D)C . 



where D is any combination of n-p letters B and Cthe remaining ones 

 so arranged that 



[B,B2...B^] = [CD]. 



7 Grassmann, Gesamelte Werke, Vol. I, p. 83; Whitehead, Universal Alge- 

 bra, p. ISS; H. B. Philips, Proceedings of the American Academy of Arts and 

 Sc'iences, Vol. 46, p. 369. In this last article the formula obtained may have a 

 different sign from the one here given. 



