2 
several coefficients of the cube, the square and 
the first power of the dependent variable, and 
also the absolute term, to vanish separately. 
These four conditions, while they reduce the 
dexter of the non-linear to zero, enable us to 
eliminate four of the indeterminate multipliers 
from its sinister. No elevation of degree will 
arise from the elimination, for all these four con- 
ditions are linear. The co-efficients of the six 
homogeneous quadratic products on the sinister 
will now in general be homogeneous linear 
functions of the two uneliminated indetermi- 
nate multipliers, and, by the solution of a cubic 
only, the ratio of these two multipliers can be 
so assigned as to cause the sinister to break up 
into linear factors, each factor being a linear 
and homogeneous function of the three elements. 
If we apply the exponential substitution, to 
either of these factors equated to zto, the re- 
sulting final non-linear differential equations of 
the first order are of a soluble form. We have 
thus constructed a soluble non-linear differential 
resolvent of a general biquadratic. For a 
cubic we might dispense with one 
of the homogeneous products, and conse- 
quently with one of the indeterminate multi- 
pliers, but we should thus be led to a resulting 
cubic, and it will be better to retain the whole 
six terms of the sinister. We shall then, 
having only three conditions of evanescence to 
satisfy on the dexter, be able to break up the 
sinister into linear factors, as before, by means 
of a homogeneous cubic in the three remaining 
disposable indeterminate multipliers. Apply- 
ing, to this last cubic, the method of 
vanishing groups, we see that its solu- 
tion depends upon the solution of a quadratic 
equation, and the extraction of a cube root only. 
In the case of a quartic, the integral obtained 
by the foregoing processes involves two arbi- 
trary constants only, and its nature and extent 
requires further discussion. But it seems that, 
by means of the theory of co-resolvents, we 
obtain new methods of solving algebraic equa- 
tions up to the fourth degree inclusive ; and, 
although the above discussion does not embrace 
equations whose degrees exceed four, it appa- 
rently indicates that further results may spring 
from the study of non-linear differential resol- 
vents. 
At the conclusion of the reading of the above 
paper by Chief Justiee Cockle, Mr. Suter read 
the following remarks, with reference to a 
paper read by him at the last meeting of the 
Philosophical Society : — 
In reference to the paper read at the last 
meeting of the society and the mention made 
therein of the new Hospital now building, 
Mr. Suter wished to say by way of explanation 
that he had purposely avoided, from motives of 
delicacy, giving any detailed account 
of the plan and arrangements, seeing 
that the building was not complete, and 
any judgment on it must therefore be 
necessarily premature. Since the meeting, 
however, he had had an opportunity of going 
over the works, and Mr. Tiffin had very kindly 
shown him the drawings and plans, and it was 
a matter of great gratification to him to find 
that the positions he had advanced were to a 
very great extent supported by Mr. Tiffin, and 
carried out by him in actual practice. The 
new hospital would, under somewhat difficult 
circumstances of position and locality, combine 
most if not all the requirements demanded by 
the increased knowledge and skill of modern 
times. The wards were spacious and well 
ventilated, and the amount of cubical super- 
ficial area per bed, if not superabundant, 
was at least adequate. By the ar- 
rangement adopted, the wards were for 
the most part broken up into detached portions, 
each forming as it were a hospital complete in 
itself, and provided with all the necessary ac- 
commodation of baths and lavatories. Thus all 
risk of contamination and infection was re- 
duced to the minimum. Had the site pre- 
sented fewer difficulties, and the resources at 
Mr. Tiffin’s command been less limbed, he 
would doubtless have preferred a different 
system of drainage, and the adoption of other 
appliances required in a hospital ; but notwith- 
standing the very few drawbacks already alluded 
to, it was very gratifying to feel that full ad- 
vantage had been taken to erect a hospital 
worthy of Brisbane, and which will prove not 
only an ornament to the oityand a credit to the 
architect, but a blessing to the people. 
Printed by G-, Wight, “ Guardian Office,” Brisbane. 
