Mr. J. Cockle's Analysis of the Theory of Equations, 499 



others. Hence we may always confine our attention to the 

 non-linear equations, and consider that as done which is always 

 possible, viz. the elimination of as many unknowns as there 

 are linear equations. But, to our question. In general, two 

 binary quadratics do not admit of solution by quadratics and 

 linears only : but such a system may be solved when either the 

 squares or the product of the unknowns appear in every term 

 involving the unknowns. It will shortly be seen that this par- 

 ticular case introduces into the theory of equations a general 

 process of solution. 



14. We have now exhausted the subject of the determinate 

 solution of quadratics by means of quadratics, and are about 

 to enter on the exhaustless field of indeterminate quadratics, 

 that is to say, of systems of simultaneous quadratics whose 

 order is greater than the number of equations to be satisfied. 

 For, true to a principle originally laid down, I shall discuss, 

 as far as practicable, the various species of quadratics before 

 going to the higher genera of equations. The first question, 

 of the indeterminate kind, that is presented to us is,— Whether 

 it be possible to solve two simultaneous tertiary quadratics 

 by means of quadratics and linears only? The answer is,— Yes, 

 by various processes, an outline of which I give below. 



(a.) They may be solved by a combination of superposition, 

 transformation, and the process used in solving two binary 

 quadratics free from the first power of the unknowns. Calling 

 this the homogeneous process, we may sketch our solution as 

 follows : — By successive superpositions (in the first of which 

 the superposed quantity involves two, in the second one, and 

 in the third none, of the unknowns) we may reduce one of the 

 given quadratics to the form of a sum of four squares, one of 

 which does not involve the unknowns. Take the linear func- 

 tions of the unknowns of which the squares are obtained by 

 these superpositions as new unknowns. Transform the second 

 of the given tertiary quadratics into another in which the new 

 unknowns shall be the only undetermined quantities. As 

 none of the old unknowns have yet been determined, so the 

 new system is as indeterminate as the original one. Now, de- 

 termine one of the three new unknowns, in terms of the other 

 two, in such a manner as that neither of the three shall enter 

 linearly into the second quadratic. Then the two given ter- 

 tiaries are reduced to two binaries of a form which, as we have 

 already seen, is completely solvible by quadratics. I term 

 this the method of homogeneous elimination, and 1 have given 

 its application to the solution of two tertiaries under a some- 

 what different form (see Mechanics' Magazine, vol. xlviii. 

 pp. 605, 606), by decomposing each unknown into the sum of 

 two new unknowns. 



2 K2 



