494 Mr. J. Cockle on the Method of Symmetric Products. 

 of the method of symmetric products that if we make 



y=f{'^)={a+x)-\ 



we arrive at the solution of a cubic which I gave at pp. 248, 249 

 of vol. ii. of the Cambridge Mathematical Journal. A sketch 

 of the process by which this solution is shown to fall under the 

 method will be 'found at pp. 228, 229 of the 52nd volume of 

 the Mechanics' Magazine. In a note to my paper on the Method 

 of Vanishing Groups published in the Cambridge and Dublin 

 Mathematical Journal tor May last, I developed the application 

 of the method of symmetric products to the solution of a biqua- 

 dratic. As my purpose is not to repeat but to endeavour to 

 extend former results, I shall content myself with referring to 

 the latter paper. 



5. There is a species of symmetric function which I have called 

 'critical,' and considered in this Journal (S. 3. vol. xxviii. p. 191), 

 and with great detail in the third and concluding volume of the 

 Mathematician. Their characteristic property, and one that has 

 an important bearing on the theoiy of equations, is that if to 

 each of the quantities (yj, yc^, . , y„ for example) symmetrically 

 involved there be added the same quantity [b), the transformed 

 function (of y-^ + b, y^ + b, . . .) h free from b, and does not differ 

 in value from the original one. Let us represent a certain 

 normal form of homogeneous critical function by <}>„{yj, where, 

 as above, ?« is the degree and n the order of the function ; and let 



and in general 



Then, if </, c^ and c.^ be certain determinable constant multipliers, 

 and, as in the preceding article, y be supposed to replace v in P, 

 the following relations hold, 



[F=c'(C')*], T,=c,C„ V,=c,C,. 



6. When n is greater than 4, can we obtain the analogous 

 relation 



P„=c„C„? 

 Or, if not, and we have 



T„-i(?/„) = <?„_iC„_i+R„_„ .... (a) 



what are the value and properties of R„_i? Or, can we attain any 

 available results by taking a value of ?n greater or less than n— 1 ? 

 If R„_i vanishes, the answer to the second of these questions 

 will give an affirmative reply to the first. And, under any cii-- 

 cumstances, there are considerations which seem to render such 

 an assumption as {a) a desirable one. 



7. Any proposed extension of the method of symmetric pro- 



