494? Mr. J. Cockle's Analysis of the Theory of Equations. 



justification of the use of this last term, I may observe that 

 Sir J. W. Lubbock has applied the term quasi-literal to cer- 

 tain astronomic developments (Theory of the Moon, Pre- 

 face to Part V., p. ix.). 



4. We thus have three great classes of equations, — (1) the 

 algebraic, (2) the quasi-algebraic, and (3) the numeric. And 

 this is no arbitrary classification, but one arising from the very 

 nature of the subject; for, call these classes by what name we 

 will, the results aimed at in the first are very different from 

 those anticipated in the second and third, which last, again, 

 have different aims and processes. The theory of algebraic 

 equations belongs exclusively to Symbolical Algebra, and 

 rigorous symbolical results are the only ones aimed at. In 

 the qua si -algebraic theory we investigate such general pro- 

 cesses, and properties of equations, as shall be ancillary to the 

 operations of the numeric theory, and shall enable us to apply 

 the latter operations with certainty and effect. 



5. Adopting substantially the arrangement given in my 

 former letter, we shall have the whole Theory of Equations 

 comprised in seven departments, of which the algebraic theory 

 contains four; — viz. I. Elimination and Solution; II. Trans- 

 formation; III. Multiplicity of Solutions; IV. Relations 

 between Coefficients and Roots, including the theory of Sym- 

 metric Functions. In the quasi- algebraic theory are comprised 

 two departments, viz. V. The nature of the roots; VI. Their 

 limits. To the numeric theory appertains VII. The nume- 

 rical values of the real roots and of the real quantities involved 

 both in the real and imaginary parts of the expressions for the 

 unreal roots. 



6. Hence, neither the quasi-algebraic nor the numeric 

 theory belong exclusively to Arithmetical Algebra. It is con- 

 ceivable that an unreal root may be completely determined 

 by purely numerical processes ; and, in fact, Dr. Rutherford 

 has, in his 6 Complete Solution of Numerical Equations,' en- 

 tered upon such a course of inquiry. To this I may add that 

 you yourself, in some notes printed with my former letter, 

 evidently regard the determination of the unreal roots as 

 coming within the scope of the numeric theory. In like man- 

 ner, the quasi-algebraic theory contemplates the existence of 

 unreal roots, and seeks to ascertain their nature and limits. 

 As my purpose is, however, not to discuss the last-mentioned 

 theories, but to proceed to that of equations purely algebraic, I 

 must dismiss the subject — at least for the present. But, before 

 I do so, I cannot refrain from naming a most important addition 

 to the quasi-algebraic theory, in the shape of a tract on 4 The 

 Analysis of Numerical Equations,' which constitutes the First 



