554 Proceedings of Royal Society of Edinburgh. [sess. 
Of course it follows from (4) that 
G(/x , sp + p) = G(p , p) 
and from (4) and (1) that 
G(p,Sp) = 0 (/*>!)• 
(11) Now p and v being any positive integers whatever and 
p t , v t the last divisor and dividend in the ordinary division-process 
of finding the greatest common measure of p and v, it is clear from 
(3) and (4) of the preceding paragraph that 
G(/x , v ) = G (p t , v t ) . 
If p and v be mutually prime, p t = 1 , and therefore G(^ , v t ) is 
by (2) of § 10 equal to 1 : if on the other hand p t >\ , G(^ , v t ) is 
by (4) of § 10 equal to G(/^ , v t ) and therefore by (1) is equal to 
0. We thus have the theorem : — 
The value of the determinant G(/x , v) is 1 or 0 according as p is 
prime to v or not. 
(12) Taking the result just obtained along with the theorem of 
§ 8 we deduce the result 
The circulant C(a , a ... , b , b) whose elements are p tis followed 
by v b’s is equal to 
(pa + vb) (a — hy+ v ~ 1 or 0 
according as p is prime to v or not. 
Putting a = 1 , b = 0 in this we obtain the result of Catalan’s 
§§ 19, 20 : and putting a= -1,6=1 we obtain the result which 
ought to have been given in Catalan’s § 18. 
(. Issued separately October 21 , 1903 .) 
