508 Mr. J. Cockle's Analysis of the Theory of Equations* 



although we may perhaps regret that this particular subject 

 did not possess greater interest to them. 



30. I do not intend to trouble you about the limits of trans- 

 formation of equations. I shall only say that possibly, nay, 

 probably, the limits of the application of all indeterminate pro- 

 cesses are the same as those imposed by Sir W. R. Hamilton 

 on Mr. Jerrard's method. But I must be allowed to add a 

 word with respect to the failure of Mr. Jerrard's solution of 

 the equation of the fifth degree by means of indeterminates, 

 and also with respect to the failure of my own solution. It 

 must not be thought that either of those solutions involve any 

 error of principle. On the contrary, the principles involved in 

 both of them may be advantageously employed in the solution 

 of many problems. The failures in question arise from the 

 root of the general equation being, in both instances, pre- 

 sented in the shape of a vanishing fraction. In both cases 

 the ball is well-directed, but it wants the impetus required to 

 carry it home ; in neither is the arrow wrongly aimed, but it 

 shivers against an invisible barrier interposed between it and 

 its object. 



31. I may here remind you that, in the third and final vo- 

 lume of the Mathematician, I have given three papers on the 

 Theory of Symmetric Functions. These papers are chiefly de- 

 voted to the application of Mr. Jerrard's theory to the subject 

 of what I have (Phil. Mag. S.3.vol.xxviii.p.l91,and elsewhere) 

 termed critical functions. In the third paper (Math., pp. 30, 

 31 of the Supplement to vol. iii.) I have accounted, d priori, 

 for the occurrence of these critical functions. It might per- 

 haps be desirable to give a further discussion of what we may 

 call the semi-critical functions ; that is to say, of functions into 

 which P enters to less dimensions than those of the function. 

 The occurrence of these critical and semi-critical functions 

 must be narrowly watched. Their occurrence will often in- 

 troduce vanishing fractions into expressions where we should 

 little have anticipated their occurrence. While upon the 

 subject of symmetric functions, I may add that I think there 

 is great weight in a remark at the conclusion of Mr. Steven- 

 son's treatise on equations (2nd edit.). I have given a spe- 

 cimen of such equations, as Mr. Stevenson probably had in 

 view when he penned the conclusion of that work, in the Phi- 

 losophical Magazine (S. 3. vol. xxvii. p. 294;). 



32. I still propose to adhere to the same use of the terms 

 direct and indirect, when applied to solutions of equations, as 

 that which I adopted in my former letter to you on the pre- 

 sent subject. But I would notice that Euler's method is at 

 the same time a direct and an indirect one. It is direct so 



