432 REPORT—1883. 
I observe that for a corpus of any order the function mn—nm is invariantive for 
any linear substitution impressed on m and . Its determinant will therefore be an 
invariant for any substitution impressed on mand x. When mand are of the second 
order, reducing each term of (mn—nm)?, i.e. mamn—mn*m—nm?n + nmnm, and of 
mn —nm, by means of the three identical equations to the form of a linear function of 
mn, m, n, 1,it will be found without difficulty that there results the identical equa- 
tion (mn—nm)?+I=0, the coefficient of mn—nm vanishing. Consequently the 
determinant of the matrix mn—mm is equal to J, which on calculation will be found 
to be identical with theinvariant of the ternary quadric function. 
It is obvious from the three identical equations that if m, n are in involution— 
that is, if their involutant is zero—every rational and integral function of m, n will 
be in involution with every other rational and integral function of m,n. Hence 
follows this new and striking theorem concerning matrices of the second order: 
If f (m,n) and ¢ (m,n) are any rational functions whatever of m, n, the deter- 
minant to the matrix mn—nm is contained as a factor in the determinant to the 
matrix fo—o f. 
[It may be noticed that f,p need not be integer functions by stipulation, because 
any linear function of mn, m,n, 1, divided anteriorly or posteriorly by a second like 
function, can itself be expressed as a linear function of the same four terms. | 
As a very simple example of the theorem, observe that the determinant of 
mn —mnm will contain as a factor the determinant of mn—nm. 
3. On a Modification of Bunsen’s Ice Calorimeter. By Professor 
Ba.rour Stewart, F.R.S. 
This instrument was exhibited to the Section. It was designed in order to 
obviate the experimental difficulties attending the use of Bunsen’s calorimeter. 
In it the tube containing water is retained as the essential part of the instrument, 
but instead of being fused into an outer vessel filled with ice it is fused into the 
bulb of a large spherical mercurial thermometer, after the manner of Fayre and 
Silbermann, so that while the inside contains water the outside is in contact with 
the mercury of the thermometer. The whole thermometer-bulb is inclosed in a 
copper envelope, which surrounds it without contact, and this copper envelope is 
kept at 0° C. by being surrounded with melting ice. Under these circumstances the 
temperature of the thermometer will also be at 0° C. This temperature is recorded 
by a very open scale, so that a small rise may be easily seen. The experiment is 
made as follows:—After the whole has been surrounded sufficiently long with 
melting ice, the substance whose specific heat we wish to find is dropped into the 
ice-cold water of the tube. Its heat is then rapidly communicated, first to the 
water, and from it through the glass of the tube to the mercury of the thermometer, 
and the rise of temperature of the latter is recorded on its stem. Theoretically a 
slight correction has to be made for the heat which is given out during the progress 
of the experiment by the (now) heated thermometer to the copper envelope which 
surrounds it; but in practice this may be disregarded, and the instrument is found 
to give us a rapid and sufficiently accurate method of determining the specific heat 
of such substances as can only be procured in small quantity. In using the 
instrument it is desirable to have a small quantity of mercury at the bottom of the 
water in the glass tube, by which means the heat is more rapidly communicated. 
4. On some Measurements of Glacier-Motion in 1883. By Professor 
Arruur Scuuster, F.R.S. 
A change has once more taken place in the general motion of Swiss glaciers, 
for they are again pushing forward into the valleys. For a year or two it had 
been noticed that their upper ends increased in bulk, and now there is no doubt 
about the steady advance. The author's personal information has been gained in the 
Chamonix valley, but the reports from different parts of Switzerland seem all to 
