in the Theory of Determinants. 425 



notation ; for this also he could have done, as may be seen 

 on looking at p. 211, &c, where he uses 



(ah') for ab'-a'b, 



(ab'c") for (ab 1 -a'b)c n -{ab' J ' -a!'b)c' + (a'b" ' -a"b')c, 



and so on. It was reserved for Professor Cay ley, a hundred 

 years after the date of Bezout's work, to point out that the 

 " proper proof" of the identities is by means of what we have 

 called the generating gnomon (Quart. Journ. of Math. xv. 

 pp. 55-57). 



Sect. I. Chap. 3. 



13. The title of this chapter is "Transformation of determi- 

 nants into other determinants when the elements are connected 

 by linear equations." What it really gives is the familiar 

 solution of a set of simultaneous linear equations by means of 

 determinants. 



Sect I. Chap. 4. 



14. The subject-matter here is a special form of determinant, 

 viz. that in which each element is zero whose column- number 

 exceeds its row-number by more than unity. The one theorem 

 given is very interesting, and has not, I think, been published 

 elsewhere up till now. 



For the purpose of more readily giving expression to it, let 

 us use the symbol 



(a b c d e; f) 2 



to stand for the sum of all the terms whose factors are / and 

 two of the letters a, b, c, d, e ; that is to say, for 



abf+ acf+ adf+ aef+ bcf+ bdf+ bef-{- cdf-\- cef+ def; 



and, similarly, 



(1; 2,3,4,5,6), 



for 



123 + 124+125 -f 126 + 134 + 135 -f 136 + 145 + 146 + 156; 



and thus finally 



/a, b,c, d, e; f\ 



Vl;2,3, 4,5, 6 7 2 

 for 



« A/a + 01^3/4 + M2/5 + «i<? 2 /6 + V3/4 + M3/5 + bie 5 f 9 



+ Cidifs + c Y e A fs + d&fe; 



the combinations of a, b, c, d, e and of 1,2, 3, 4, 5, 6 taken 

 two together being arranged in order. Further, let us spe- 

 cialize the form of the determinant somewhat more by making 

 Phil. Mag. S. 5. Vol. 18. No. 114. Nov. 1884. 2 F 



