Note on Bosanes' Functions Bcsemhling Jacohians. 161 
10. The various identities — fifteen in number — which have just been 
estabUshed, may be conveniently arranged thus : — 
The last twelve are arranged in pairs, each one differing from the one 
over against it in having u,v in place of v,u respectively. The first three 
have no such counterparts, being unaltered by the interchange in question. 
Let it be noted also that six of them (Nos. 1, 4, 5, 6, 7, 8) are obtained 
as evanescent determinants, and the remaining nine as deductions there- 
from after differentiation. 
An attempt to obtain more results of the like kind will be found 
unfruitful: it will, however, be none the less interesting and instructive. 
11. Incidentally in § 9 we have in effect established Rosanes' theorem ; 
for the results there obtained, when written without the help of the 2. 
are the equations 
which, when we solve for the give 
111 '■ : : 7/^ = u\ : u\ : u\: /r^. 
13 
