ESSAYS 131 



arising in the Theory of Numbers, and in particular from Riemann's celebrated 

 researches on Prime Numbers. 



The present age may see, or perhaps is seeing, a revival in Great Britain of 

 interest in such parts of mathematics as have marked or are likely to mark a 

 definite advance of the science. This would be greatly facilitated if lecturers 

 were to lecture more frequently upon other than the ordinary examination subjects. 

 The Universities might make it easier for qualified members of the public to 

 attend such lectures and might, perhaps, confer upon them some sign of recogni- 

 tion after a definite period of work. It is no easy matter to decide upon what the 

 recognition should depend, but written examinations in very advanced work are 

 by no means a test of proficiency. Proficiency is really indicated by a general 

 knowledge of the principles of an investigation and not necessarily by the ability 

 to write out in three hours the details, which one is often only too pleased to take 

 for granted when reading books or journals. 



At present most of the advanced students in the Universities are those who 

 have taken the usual undergraduate course. It would contribute greatly to the 

 progress of the science in Britain and of the Universities if some attempt were 

 made to attract those students who find their requirements satisfied only at 

 continental Universities. 



ME. J. H. GURNEY'S SOLUTION OP QUARTIC EQUATIONS 



(A. S. Percival, M.A., M.B. Camb.) 



When the coefficients of the Quartic are numerical, Horner's tedious method is 

 commonly used. For some little time I have been replacing the Quartic by two 

 quadratic factors. This can be done in several ways, and I ha^e been corre- 

 sponding with Mr. Gurney about the simplest method of forming these factors. 

 He finally has given this method, which I feel convinced is the speediest way of 

 finding all the four roots of the equation. 

 Let the Quartic be of the form 



x k + Ax 3 + Bx* + Cx + D = o or F(x). 



If A X o, remove the second term, synthetically dividing by . In other words 



4 



transform F(x) into/(_y) by putting x = y ; we then have 



(0 /O0=y+ Qf + Ry + S = o. 



(2) Write down the auxiliary cubic <fi{z 3 ), where 



£(*») - z 6 + 2QJ> + (Q 2 - 45>* - R* = o. 

 This cubic must have one positive root at least. Call it a*. 

 Then 4>te J ) = (* 2 - a') (z* + pz i + q). 



(3) It will then be found that 



f{y)~ {y +ay + i(a' +p-2sjq)}{f - ay + i(a* +j> + 2^)} 



and hence the four roots of the Quartic are found by solving these two quadratics. 

 It should be noted that the positive values of \/a 2 and *Jq must be taken 

 Further that 



(1) If the cubic has 3 real positive roots, the quartic has 4 real roots. 



(2) If the cubic has 1 real positive root and 2 unreal roots, 



the quartic has 2 real and 2 unreal roots. 



