2 
't 
general linear differential equation of the second 
order is absolutely insoluble by any finite for- 
mula involving only algebraical, exponential, 
logarithmic, or trigonometrical expressions, or 
indeed any expressions whatever capable of 
being derived by indehnice integration from 
algeoraioal or exponential functions : — and tnis 
impossibility subsists even although the derived 
functions be supposed to be affected in any 
way whatever, and any Unite number ot ways 
whatever, by signs of indefinite integration. 
The demonstration of the impossibility of 
solving the general linear differential eqnaiion 
of the second order I arrive at by pushing one 
Step further the reduction of equation (148) of 
art. 77 of my “ Notes on the Differential 
Calculus,” in vol. iii. of the Messenger of 
Mathematics. On expunging the traces of 
the independent variable (128) takes a lorm j 
the solution of which is impossible. (2.) Tne 
second and third terms of a linear differential 
equation of the third order can be simultane- 
ously destroyed if we assume the solubility of 
a linear differential equation of the second order* 
If we simultaneously change the dependent and 
independent variables, form the condiiions, and 
eliminate properly, we shall be le 1 to a linear 
differential equation of the second order, in 
which the depeudent variable is the square of 
the modulus (or factor) in the factorial sub- 
stitution by which the dependent variable is 
supposed to be changed. (3.) The class of 
aigeoraical equations which lead to homogene- 
ous differential resolvents are nob necessarily 
wanting in their second term : the resolvent is 
homogeneous it each root is a linear and homo- 
geneous function of other roots. (4.) I have 
arrived at various new, as I believe, and com- 
paratively general forms of soluble linear dif- 
ferential equaiions of the second order. 
The Chief Justice added that other engage- 
ments or occupations had prevented him from 
drawing up any formal paper embodying the 
above results. But he hoped that, in a scientific 
point of view, they -would be found of sufficient 
interest or importance to justify him in making 
the foregoing announcement to the society. 
Briuted by Q, Wight, “Guardian Office,” Brisbane, 
