Mr. J. Cockle’s Note on the Remarks of Mr. Jerrard. 331 
amounted to 44°1 grains in an imperial gallon, of which by far 
the largest proportion consisted of earbonate of soda. 
The alkalinity of the boiled water was determined by means 
of standard sulphuric acid, and found to be equivalent to a pro- 
portion of 30-76 grains of carbonate of soda in an imperial gallon. 
The result obtained by the direct determination of the carbonic 
acid, corresponded accurately to the proportion required by theory 
to hold in solution the whole of the lme and magnesia in the 
water, and to form bicarbonate with the amount of soda repre- 
sented by the number above quoted. 
The following statement represents the proportions of the 
various constituents existing in solution in an imperial gallon of 
the water :— 
Bicarbonate of soda. . . . 48°53 
Sulphate of soda. . . . - 7:90 
Chloride of sodium . . . . 1°34 
Sulphate of potassa. . . . O'dl 
Phosphate of lime . . . -_ trace 
Carbonate of lime . . . . 1°90 
Carbonate of magnesia. . . 0:80 
Organic matter . . . - - 120 
Carbonic acid, holding the carbonates of lime and magnesia in 
solution, 1:25 grain = 2°642 cubic inches at 60° F. 
The absence of nitric acid, ammonia, silicic acid, alkaline sul- 
phides, and oxide of iron was established by special examinations. 
XLV. Note on the Remarks of Mr. Jerrard, 
By James Cockz, Esq,* 
HE inverses of the rational functions, say R, by which one 
of two similar functions is expressed in terms of the other 
are themselves rational, and the inverses of those by which one 
root of an irreducible equation is (if so expressible) expressed 
vationally in terms of another are also themselves rational. And 
if B, @ be similar functions of which &,, 0, and &,, 4, are cor- 
responding values, and if moreover & be the root of an irredu- 
cible equation one root of which, &,, is a rational function, say 
r, of another, &,, we find 
6,=RE,=Rre,=RrR-6,, 
in other words, that @, is a rational function of 6,. Conse- 
quently if the equation in @ is not an Abelian, neither is the 
equation in = an Abelian. 
Again: if a rational equation be reducible, any rational trans- 
formation involving only one root gives rise to a reducible trans- 
* Communicated by the Author. 
