tfti 



MR EDWARD SANG ON THE EXTENSION OF BROUNCKER S METHOD 



eliminating^ and q from these three equations, we obtain, omitting the sub- 

 scribed „ for shortness, 



r n { A B 2 C, - A B, C 2 + B A 3 C 2 - B A 2 C x + C A 2 B 1 - C A x B 2 } = 

 {A 3 B 2 C x - A 3 B x C 2 + B 3 A, C 2 - B 3 A 2 C, + C 3 A 2 B x - C 3 A t B 2 }. 



Now the first of these quantities within the ties is the determinant obtained 

 from the coefficients 



A 

 B 



a 



A, 

 B, 



A c 





A, 



A 2 



B 



B x B 2 



C 



Cx c 2 



mi 



nant from the 





A 2 





A, 





B 2 



= 



Bx 





C 2 





Cx 



while the second is the determinant from the next three sets of values ; or, 

 using Cayley's notation 



B 2 

 C 



A 3 

 B, 



Now, in the usual operation, and when the three magnitudes are incommensur- 

 able, r is unit all along ; wherefore the determinant from nine contiguous values 

 never changes. But at the beginning this determinant is obviously unit, and 

 thus we have the ordinary well-known theorem for the usual progression in 

 reference to two magnitudes extended to the case of three ; that is to say, the 

 value of the determinant is -I- 1 all along. In the case of two magnitudes, the 

 value is alternately + 1 and — 1 ; whereas, in the case of three magnitudes, 

 the sign is preserved. 



The above statement holds good in the case of incommensurable quantities ; 

 but when there is a common measure the quotient r may disappear toward the 

 end of the operation, and then all the subsequent determinants become zero. 



If, in the case of three commensurables, we complete the calculation, as in 

 the following example in which the three primes 99137, 30763, and 3229 are 

 compared, a remarkable yet obvious law is seen. 



1 



1 



1 



1 



1 



1 



1 









r 



2 



4 



















2 











Q 



3 



9 



8 



2 



1 



1 



3 



11 



3 





P 

 A 



1 



3 



29 



245 



493 



522 



767 



2794 



32790 



99137 





1 



9 



76 



153 



162 



238 



867 



10175 



30763 



B 







1 



8 



16 



17 



25 



91 



1068 



3229 



C 









1 



2 



2 



3 



11 



129 



390 



D 











1 



1 



1 



4 



47 



142 



E 



i 











1 



1 

 1 



3 

 3 

 1 



36 



35 



11 



1 



109 



106 



33 



3 



1 



F 



G 



H 



I 



K 



