310 



LIPKA. 



and by Ciu, ir its corresponding co-minor of the (n — ?,) d order, and 

 expand by Laplace's theorem ^^ in terms of the (n — 1) (n — 2)/2 

 minors of the 2d order formed from the columns headed cn and cn, 

 we get 



(66) qm= ( — 1)" [CtlW2 C'ij,i2 + Cii,/3 Cii,i3+ . . . . + Ci„_2,J„-l 



^tn-2)ln-lj> 



where v is the sum of the columns in pm headed by cn and cn. 



From the matrix (62), we see that the determinants qm, Qi, and qi are 

 formed by omitting the columns headed Cmi, cn, and cn, respectively, 

 and that, therefore, the minors of the 2d order 



Clk, 



^ Ik (^mk 

 (' Ir Cjn J- 



and Cik, mr — 



Cik Cmk 

 Cir Cm.T 



in the expansions for qi and qi, respectively, have the same co-minors 

 of the {n — 3) d order that the minors of 2d order, Cik, ir, have in the 

 expansion for qm. Designating these common co-minors by C12, 

 Ci3, . . . ., C„_2, „-i, we may now write the expansions 



(67) { 



qm =( — 1)' + ' [Cihn Cn + C,l,i3 C'i3+. ■ ■ ■ + Cin-2,ln-l 



Cn-2) n-lj 



qi =(— l)' + ('"-l) [CiUmiCn-i- Cii,,n3 C13+ +Cin-2,mn-l 



(-'n-2, n-lj 



qi = (- l)('-i) +(^"-1) [cn, ml C12+ Cn, mS C13+ .... + Cl „_2, ,„ „-l 



t'n-2) n-ll 



On the other hand, if the minors of the 2d order formed from two 

 columns which are not headed by two of the three terms c^i, cn, cn, 

 are multiplied by the above co-minors of the {n — 3)d order, Cn, Cu, 

 . . . . , C„_2, n-i, and added, we shall have the expansion of a determi- 

 nant with two columns alike, and hence the sum will be equal to zero. 



Let us apply this discussion to the equations (61), involving the 

 minors of the 2d order. If we expand these equations, and multiply 

 the (n — 1) (n — 2)/2 equations by the above {71 — 1) (n — 2)/2 

 co-minors of the (ti — 3) d order, Cn, C13, . . . ., Cn-2, n-i, in succes- 

 sion, and then add, we shall evidently get three terms only — all the 

 sums will vanish except those giving qm, qi, and qi. Equations (61) 

 thus assume the form 



13 See R. F. Scott, A treatise on the theory of determinants, 1880, chapt. III. 



