1887.] tts application to the solution of some equations. 107 
which is satisfied by the equation 
y (m) aa v (n2), 
or cos (m) = cos (na) + NY 2ilesin (nx) 
an equation which can hold in every case where sin(nz)=0, or 
na=0, or =7, or = 27, or =37, &c.: but which, as regards the 
concurrent value of m, holds only when na=0, or = 2z, or 
m m 2 m 4 
= 4c, &e.; or when = =a, or — =a+—, is 
n n n n n 
The remaining steps are the same as those in Professor Adams’ 
Memoir. 
(2) Some observations on Permanganic Acid. By T. H. 
HASTERFIELD. 
WHEN potassium permanganate is dissolved in strong sul- 
phuric acid, a green liquid is obtained, which, as is well known, is 
possessed of considerable oxidising power, and is decomposed with 
explosion when the temperature is slightly raised. ‘Terreil, by 
using monohydrated sulphuric acid, H,SO,.H,O,, and distilling the 
liquid so obtained upon the water bath, at a temperature of 50°— 
70°, succeeded in obtaining purple vapours; these vapours con- 
densed in the receiver to a dark crystalline mass, soluble in 
water, violently decomposed by a sudden elevation of temperature, 
and supposed by Terreil to be pure permanganic acid, HMnO, or 
H,Mn,O,. The solution of this body in water is of the same 
colour as a solution of potassium permanganate, and some chemists 
have supposed that the crystalline distillate was permanganic 
anhydride Mn,O,, the solution being an aqueous solution of per- 
manganic acid. As far as we are aware, however, no analyses of 
the crystalline distillate have as yet been published; partly no 
doubt from the danger attending its preparation. 
The primary object of the following experiments was to deter- 
mine the density of the purple vapour above mentioned; but 
though this object was not attained, the facts which came under 
notice appeared of sufficient interest for publication. 
It was suggested by Mr H. J. H. Fenton (to whose aid through- 
out my experiments I am greatly indebted, and from whom came 
the proposition of the Vapour Density determination), that the 
anhydrides of phosphoric or other acids might liberate permanganic 
anhydride from potassic permanganate. The following experi- 
ments were accordingly made :— 
(i) Well powdered potassium permanganate was dried for 
several hours at 120° in an air bath. A very slight amount of 
