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



(b.) If, after reducing one of the two given tertiaries to the 

 form of four squares as above suggested, we group them two 

 and two, and equate each group to zero; then, each group 

 being the sum or difference of two squares, is capable of being 

 decomposed into linear factors. Equate to zero one factor of 

 each group, then the tertiary so grouped vanishes. And, eli- 

 minating, by means of the two vanishing factors, two of the 

 unknowns from the remaining tertiary, the latter becomes a 

 simple quadratic in the third unknown, which is then deter- 

 mined ; and afterwards the others are obtained by substitution 

 for the third in the vanishing factors, which are then treated 

 as ordinary binary linears. This is the method of vanishing 

 groups, of which I have given an "Account" in the Philoso- 

 phical Magazine (S. 3. vol. xxxii. pp. 114-119), and to which, 

 as far as I had then developed it in print, you will find refer- 

 ences in a note [J] to my fifth paper on the Transformation of 

 Algebraic Equations, at pp. 178, 179 of the third and conclu- 

 ding volume of the Mathematician. 



(c.) Although Mr. G. B. Jerrard's method of vanishing co- 

 efficients (as I have proposed to term it) be not directly appli- 

 cable to the solution, in their explicit form, of three tertiary 

 quadratics by quadratics and linears only, yet, in all probabi- 

 lity, it would not be difficult, by decomposing the unknowns, 

 or some of them, into the sums of others, to give the tertiaries 

 such a form as to admit of the application of Mr. Jerrard's 

 processes. Full information respecting Mr. Jerrard's analysis 

 will be found in his ( Mathematical Researches,' and in Sir 

 W. R. Hamilton's ' Inquiry ' into the subject at pp. 295- 

 348 of the Sixth Report of the British Association. Mr. Jer- 

 rard's method of elimination differs from the ordinary one in 

 this, — that from an expression he expels, successively, all the 

 powers of a quantity, which he wishes to remain undetermined 

 in value, by making the coefficient of each power vanish. This 

 he does by a powerful algebraic indeterminate analysis, quite 

 distinct from the analysis known by that name in the ordinary 

 treatises on algebra. Those who delight to trace the analo- 

 gies which the different branches of all sciences suggest, may 

 derive satisfaction from comparing Mr. Jerrard's method with 

 that pursued in the strictly Diophantine Analysis, — that given, 

 for instance, at pp. 306-343 of the fourth edition of Professor 

 J. R. Young's Algebra. Consider, by way of example, the 

 manner of rendering rational the cube root of a cubic function 

 of x. This rationalization, when possible, is effected by ob- 

 taining an equation in x, from which all powers of x, except 

 the first, are expelled. 



15. Of course we may readily solve two quaternary qua- 

 dratics without equations higher than quadratics. For we 



