188 



DR THOMAS MUIR ON 



1 



a 



A 



aA 



1 



b 



B 



6B 



1 



c 



C 



cC 



1 



d 



D 



dD 



1 



a 



A 2 



aA 2 



1 



b 



B 2 



6B 2 



1 



c 



C 2 



cC 2 



1 



d 



D 2 



^D 2 



+ ABC 2 + . 



. + 2ABCD) - 



1 



a A 4 



«A 4 







1 



b B 4 



5B 4 







1 



c C 4 



cC 4 







1 



d D 4 



dD 4 



where 



M*= 2AB 2 C 2 + . . . + A 3 B 2 + . . . + A 3 BC + . . . + 3A 2 BCD + . . . , 

 N = (A + B + C-f-D)(A 2 + B 2 + C 2 + D 2 ) + (ABC+BCD + CDA+DAB), 

 P = A + B + C + D; 



and in connection with the denominator for the case of four arcs 



(A + B)(A + C)(A + D)(B+C)(B + D)(C + D) 



(A 2 B 2 + 



For convenience of reference these four identities may be denoted by (N 3 ), (D 3 ), 

 (N 4 ), (D 4 ) respectively. No proof is given of them by Cayley, and after stating them 

 he adds " mais je n'ai pas encore trouve" la loi generale de ces Equations." 



The object of the present paper is to do something to supply these wants. 



(2) In all the identities the determinants are seen to be multiplied by symmetric 

 functions of as many letters as the determinants have rows or columns. A general 

 theorem for the performance of such multiplications is thus seen to be desirable, and 

 the following has been found. It is a generalisation of a theorem given in 1879 in the 

 Transactions Roy. Soc. Edinburgh, xxix. p. 53. 



The product of a determinant of the n th order by a single symmetric function of n 

 quantities a, /3, y, . . . is equal to the sum of as many determinants as there o,re terms 

 in the function, each determinant of the sum being got from the given determinant and 

 a term a x fiyy z . . . of the function by multiplying each element of the first row (or 

 column) of the given determinant by the x th power of the corresponding one of the n 

 quantities, each element of the second row (or column) by the y th power of the corre- 

 sponding one of the n quantities, and so on. 



For example, 



+ 





a x a, a 

 h \ I 



c. C„ l 



3 



3 



( A 2 B + A 2 C + B 2 C + B 2 A + C 2 A + C 2 B) 



I z 



a x A 2 a 2 A a 3 

 &1B 2 Z> 2 B b 3 



C 1 C2 C 2 C C 3 



'3 



+ 



a x A} a 2 a 3 A 

 ^B 2 b 2 b 3 B 

 C-y \j Co c 3 L> 



+ 



a x a 2 A 2 a 3 A 

 \ 6 2 B 2 b 3 B 



C l C 2^ C 3^ 



a x A 

 & X B 



C,C 



a 2 A 2 a 3 

 & 2 B 2 b 3 

 c 2 C 2 c s 





4 



«jA a 2 a 3 A 2 

 b x B b 2 b 3 B 2 

 c x C c 2 c 3 C 2 



+ 



a x a 2 A a 3 A 2 

 \ b 2 B b 3 B 2 



C, Ca \J Cg \J 



The essence of the proof lies in the fact that if we single out for consideration 



* These cofactors are incorrectly printed both in the original journal and in the collection, and unfortunately 

 the mistake consists in putting small letters in place of capitals. 



