TRANSACTIONS OF THE SECTIONS. 



these papers would generously permit photographic copies to be taken. The na- 

 tional manuscripts of England and Scotland have already been admirablj' photo- 

 zincographed by Sir Henry James. The manuscripts of NeTi^ton, which are also 

 national, certainly deserve to be transmitted to posterity in like manner. 



Mathematics. 



Oil the Inverse ProJ^lem of Co resolvents. By the Hon. J. Cocele, M.A., F.E-./S. 

 Communicated hij the Eov. Professor 11. Haklet, F.R.S. 

 Inverse problems, as is well known, present greater difficulties than direct ones. 

 For instance, while it is easj' to square a niunber, it is not so easy to extract its 

 square root. Moreover, there are cases in which it is impossible to obtain a finite so- 

 lution of an inverse problem. The solution of a quintic is usually considered to be 

 such a case. In the theory of coresolvents it is comparatively easy to pass from 

 the algebraical to the differential resolvent, but the converse does not hold. The 

 finite integration of the linear diil'erential resolvent of a given algebraical equation 

 would, perhaps, be a step towards the general solution of the inverse problem. 

 But that integration has not yet been effected, except in two or three special cases ; 

 and the definite integrals of Boole have not, that I am aware of, been converted into 

 indefinite ones. In order to take the step abo\-e pointed to, it seems to me neces- 

 sary to have recourse to a non-linear differential resolvent, to be constructed as 

 follows: — The elements of the final non-lineav are three; the first is (l)the second 

 differential coefBcient of the dependent variable ; the second is (2) the first 

 differential coefficient of that variable ; the third is (.3) the square of the second 

 element divided by the dependent variable itself. The sinister of the non-linear 

 resolvent is constituted by the six homogeneous quadratic products of the three 

 elements, and is the sum of those six products, each multiplied into an indeter- 

 minate or conditional multiplier. Each element and each product is, as we know 

 by the theory of coresolvents, in general capable of being expressed as a rational 

 and integi'al function of the dependent variable, of a degree less by one than that of 

 the given algebraic equation. Suppose this last equation to be a quartic, then 

 each product, and consequently the dexter of the non-linear resolvent, can be ex- 

 pressed as a culjic function of the dependent variahl(>. Let the dexter of the non- 

 linear be reduced to zero by causing the 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 condi- 

 tions are linear. The coefficients of the six homogeneous quadratic products on 

 the sinister will now in general be homogeneous linear functions of the two un- 

 eiiminated, indeterminate multipliers ; and, by the solution of a culjic only, the 

 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 tho 

 three elements. If we ajjply the exponential substitution to either of these factors 

 equated to zero, the resulting final non-linear difl'erential 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 consequently v,-ith one of the indeter- 

 minate multipliers ; but we should thus be led to a resulting cubic ; and it will be 

 better to retain tho 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 cidjic in the three 

 remaining disposable indeterminate nmltipliers. Applying to this last cubic the 

 method of vanishing groups, we see that its solution depends upon the solution 

 of a quadratic equation and the extraction of a cube root only. In the case of a 

 quartic, the integral obt.ained by the foregoing processes involves two arbitrary 

 constants onlv, and its nature and extent require further discussion. But it seems 



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