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



vanish at a critical stage of our superpositions, the general 

 method admits of modification, I have exhibited a particular, 

 but useful, case of such modification (I allude to the case of 

 tertiary functions) at page 547 of vol. xlvii. of the Mechanics' 

 Magazine. In the case where some or all the squares disap- 

 pear from a tertiary quadratic function, Professor Hearn has 

 (Mathematician, vol. iii. p. 198) suggested a process which 

 may also be generalized and adopted in place of my modifi- 

 cation. On Professor Hearn's neat and able discussion of 

 surfaces of the second degree I have made short comments at 

 p. 249 of vol. iii. of the Mathematician, and also at pp. 44, 45 

 of the Supplement to that volume, with which (supplement) the 

 existence of the valuable periodical in question terminated. 



17. Having sufficiently discussed the quadratic solution, I 

 proceed to the biquadratic determination of complex quadra- 

 tics, by which I mean such a determination of them as shall 

 render their solution possible, without having to satisfy any 

 simple equation of a degree higher than a biquadratic. Now 

 there is one system of three tertiary quadratics, the solution 

 of which is capable of being rendered dependent on that of 

 a simple biquadratic. I mean the case in which all the three 

 given equations are, what I shall call, pseudo- homogeneous, that 

 is to say, are free from the linear dimension of x. The system 

 of equations involved in " Colonel Silas Titus's Problem " 

 belongs to such a system. I have given a discussion, and 

 some references relating to this problem, at pp. 34-36 of vol. 1. 

 of the Mechanics' Magazine. The method by which I have 

 solved three general pseudo-homogeneous quadratics (Mech. 

 Mag., vol. xlviii. p. 512) is, in its principle, the same as that 

 by which Frend solved Titus's problem. And Frend's me- 

 thod is an ingenious and philosophical generalization of the 

 process employed in solving two pseudo-homogeneous binary 

 quadratics, unless indeed the latter is a particular case of 

 Frend's. In the form in which I have exhibited the solution 

 of pseudo-homogeneous binary (See Mech. Mag., vol. xlviii. 

 p. 511) and tertiary quadratics, we avoid the introduction of 

 new unknowns. 



18. If we had three general quadratics of the fifth order, 

 we might effect their biquadratic determination by the method 

 of homogeneous elimination (Ibid. p. 606) ; but then we must 

 decompose each of the unknowns into the sum of two others. 

 The same determination may be effected, without any decom- 

 position, by the method of vanishing groups. Thus, if, by 

 five superpositions, we reduce one of the quadratics to the form 

 of a sum of six squares, we may group these squares two and 

 two, make the three sums vanish, depress, and eliminate three 



