648 
Proceedings of Royal Society of Edinburgh . [sess. 
The Theory of Continuants in the Historical Order of 
Development up to 1880. 
By Thomas Muir, LL.D. 
(MS. received December 12, 1904. Read January 9, 1905.) 
The first of the writings dealt with in the present communica- 
tion is of a considerably earlier date than the others : its proper 
place is immediately in front of Sylvester’s paper of the same 
year. The account of it here given should therefore be found 
between pages 140 and 141 of this volume of the Proceedings. 
Smith, H. J. [S.] (1854, May). 
[De compositione numerorum primorum formae 4A. 4- 1 ex duobus 
quadratis. Crellds Journal , 1. pp. 91-92.] 
Pointing out, as Sylvester had already done, that the writing of 
Euler’s algorithm (q 1 , q 2 , . . . , qf) as a determinant leads easily to 
the properties 
(?!, 0 2? ■ • • ^ = 2«- n • • •> 0i) 5 
fel , 02 3 • < • 3 0») = (01 > 02 » • • • I 0i) (0i+ 1 > • • • 0«) 
+ (01 , 02 > • • • ) 0<-l) fe+2 3 • • • 3 0*) , 
Smith therefrom deduces that 
and 
(01 3 02 5 • * • j 0i— 1 > 0t J 0i+l 5 • • • 3 0l) ~ (01 3 02 3 • • • 3 0i-l) j 0], 3 • • • 3 0i) + (01 ) • • • 5 0i-2) 
noting in regard to the former that the two numbers squared on 
the right are mutually prime. He then makes application to the 
theorem referred to in the title of his paper. 
