470 Mr, A. Caylcy on a Question in the Theory of Probabilities. 



very little reflection, however, will convince us that the question 

 has a far wider scope than this answer would imply. 



13. Doubtless Si ought to vanish ; in other words, Sought 

 to be such as to become equal to zero for those particular values 

 of A u A^, . . A 5 which lead to a solvible case involving fifth roots. 

 If, then, ® be so constituted as to apprise us of the existence of 

 all such solvible cases — as it unquestionably ought to do — it 

 must involve factors by the evanescence of which each expression 

 for 5)| shall vanish. Accordingly S must not be composed 

 merely of a succession of terms of the form x A a B* 3 C y . . E e i 

 A, B, C, . . E having the same meanings as A v Ag, A 3 , . . A 5 

 respectively. Now this function ®, in which A, B, C, . . E are 

 all of them supposed to be arbitrary, cannot vanish unless — in 

 opposition to what has been just stated — it be made up of 

 terms of the form x A a B^ C y . . E e . I conclude therefore that 

 2) ought to be different from zero. 



14. Again, when I regard 2) from another point of view as 

 the denominator of an expression indicative of an impossibility, 

 X am forced to come to the conclusion that S ought to be equal 

 to zero. 



15. Reflecting on the incongruous properties which are thus 

 seen to attach themselves to the function S, I am unable to 

 accept the result 



N 



2> 



as free from error. 



16. In my next paper I purpose to meet some objections urged 

 by Mr. Cayley and Mr. Cockle against my proof, in the Philo- 

 sophical Magazine for May 1861, of the impossibility of establish- 

 ing a rational communication* between the function fifPffiffi 

 and its fifth power. 



March 1862. 



LXVI. Postscript to the Paper "On a Question in the Theory of 

 Probabilities " in the May Number. By A. Cayley, Esq.f 



I UNACCOUNTABLY did not recall to myself Mr. H. TTil- 

 braham's paper " On the Theory of Chances developed in 

 Prof. Boole's ' Laws of Thought/ " Phil. Mag. vol.vii.pp. 465-476 

 (1854), which contains a most valuable discussion of the ques- 



* In addition, be it remembered, to the one 



v=u 5 , 

 which characterizes the case in question, 

 t Communicated by the Author. 



