36 



NA TURE 



[Nov. 13, 1 < 



lines of a memoir by M. Darboux on the solution of a 

 biquadratic equation in Liouville's journal, with which 

 its eminent author, my colleague in the Institute of 

 France, providentially presented me shortly after my 

 arrival in Paris, and which led me to see that the three 

 pairs of solutions of the Hamiltonian equation must stand 

 in immediate conceptual relation to the three pairs of 

 sides of the complete quadrangle formed by a certain 

 conic related to the form p x' : + q x + r (in fact the deter- 

 minant of the matrix p 11 -\- q v -r r w) with the fixed conic 

 v- — u it). 



Now comes the turning point, the dvayvapitns of this 

 strange eventful history. 



" There's a Divinity that shapes our ends, 

 Rough-hew them how we will." : 

 Seized with a sudden and fortunate attack of bronchitis, 

 which confined me to my bed, and in the access of noc- 

 turnal fever which that state induces, my thoughts reverted 

 with increased activity to this geometrical figure. It 

 became clear to my inner sense that there ought to be 

 an immediate relation between the biquadratic deter- 

 minant of the form p x" + q x + r, spoken of above, and 

 the three pairs of its roots, and seizing my courage with 

 both hands, I made bold to declare to myself that the 

 functional parts of the six identical equations to the six 

 roots ought to be the three pairs of conjugate quadratic 

 factors of the biquadratic in question. 



But if this should turn out to be true, it became impos- 

 sible not to suspect, or even more than half believe, that 

 an analogous statement must admit of being made for a 

 unilateral equation {i.e. one in which, as in Hamilton's, 

 the multipliers of each power of the unknown matrix x lie 

 all on the same side (whether to the right or left) of it) 

 whatever might be the degree of the equation, and what- 

 ever the order of the matrices concerned. In other words, 

 supposing fx = o to be such equation, and </>.r = o to 

 be the identical equation to any one of its roots, (p x 

 ought to be contained as an algebraical factor in the 

 determinant of the matrix fx when, for the moment, x 

 therein is regarded as an ordinary quantity. If this were 

 so, then the reciprocal theorem would necessarily be true 

 (on account of the determinant referred to being in general 

 irreducible), viz. that, supposing w to be the order of the 

 matrices concerned, every algebraical divisor of it, say <f> x, 

 of the degree o>, must be the identically-zero function to one 

 or the other of the matrices x which satisfy the equation 

 fx = o, and consequently it would be only necessary to 

 combine, according to a well-known method of elimina- 

 tion, the given form fx with each in succession of the 

 derived forms, which constitute a brood or litter as it were, 

 issuing " de son propre sein," to obtain all the roots of fx 

 by solving the ordinary algebraical equation det. (fx) = o, 

 and that thus the solution of the unilateral equation would 

 depend on the solution of an ordinary equation of the de- 

 gree n <d, u being the degree of/in x, and 01 the order of the 

 matrices concerned : the number of the roots oifx would 

 therefore be the number of ways of combining «o> things 



o) and a together, i.e. ) — — — ^ . But herein arose a 



6 ' n<»n(« - i)<» 



self-created difficulty, a phantasmal projection of my 

 own brain, to block up the way, and throw doubt and 

 discredit on all that precedes. Supposing w = 2, the number 



of roots thus ascertained would be - - - , or iii- — ft, 



and for u = 3 would be 15. Now, in the London and 

 Edinburgh Philosophical Magazine for May last, whilst 

 I had shown that iir — n is the number of roots of 

 a unilateral equation in quaternions of the degree n, and 

 of the trinomial or Jerrardian form, I thought I had proved 

 the number of solutions of a complete cubic equation in 

 quaternions to be 21 (upon which I based the formula 



// (//' - // + 1) for a unilateral equation of quaternions of 

 the degree 11. There was then the choice to be made 

 — to abandon the conjectural theorem, or to admit an 

 error in the supposed determination of the number 21. I 

 felt no hesitation in making my election, especially as 

 there was a loop-hole for error in such numerical determi- 

 nation, inasmuch as no actual arithmetical calculations had 

 been made, but the order of a certain system of equations 

 which ought to be equal to the number of roots oifx was 

 inferred from calculations in which all numerical quantities 

 were left in blank ; it was therefore quite possible (how- 

 ever unexpected the fact) that some of the leading 

 coefficients of the resolving equation of the degree 21 

 might become zero, 1 and consequently that the order 

 might fall below (although it could not rise above) that 

 number. To my gratified surprise my faith met with its 

 reward, for I soon found an easy proof of the remarkable 

 theorem which I have ventured, in emulation of a phrase 

 of Cauchy, to call a " pulcherrima regula," which will 

 appear in the number of the Comptcs Rendus next forth- 

 coming after this date, and which may be summed up 

 approximately in the following words : — Every latent root 

 of every loot of a given unilateral function in matrices of 

 any order, is an algebraical root of the determinant of that 

 function taken as if the unknown -were an ordinary 

 quantity, and conversely every algebraical root of the 

 determinant so taken is a latent root of one of the roots of 

 the given function.- This constitutes a marvellous exten- 

 sion (to a matrix implicitly given by a unilateral equa- 

 tion) of the already no-little-marvellous Hamilton-Cayley 

 theorem of the identical equation to a matrix given 

 explicitly. 



My good genius met me on the deck of the Invicta, and 

 only left me three weeks later on board the returning 

 steamer from Boulogne. There my pleasing algebraical 

 dream came to an end. 3 J. J. SYLVESTER 



New College, Oxford, October 26 



1 It is one of Descartes' " self-evident primary t 

 has happened could not have happened otherwise. 



jths " that nothing which 



OUR FUTURE WATCHES AND CLOCKS 

 T I OWEVER long the use of the letters "a.m." and 

 ■'■•'• " p.m." for distinguishing the two halves of the 

 civil day may survive, it seems probable that the more 

 rational method of counting the hours of the day con- 

 tinuously from midnight through twenty-four hours to the 

 midnight following may before long come into use for a 

 variety of purposes for which it is well adapted, even if it 

 should not yet be generally employed. It seems proper, 

 therefore, to consider in what way ordinary watches and 

 clocks could be best accommodated to such a change in 

 the mode of reckoning. To place twenty-four hours on 

 one circle round the dial instead of twelve hours as at 

 present seems the most natural change to make ; but, in 

 addition to a new dial, it would involve also some alter- 

 ation in construction, since the hour hand would have to 

 make one revolution only in the twenty-four hours instead 

 of two. And there would be this further disadvantage, 

 that the hours being more crowded together, the angular 

 motion of the hand in moving through the space corre- 

 sponding to one hour would be less — in fact, one-half 

 of its present amount. It is to be remembered that, 

 in taking time from a clock, persons probably pay 

 small attention to the figures, either those for hours 



1 Or else that its functional part might he composite and throw off an 

 irrelevant factor. 



~ 1 11 terms more precise as regards the converse the theorem runs as follows : 

 — The identically-zero Junction to a root off* is a factor of the determinant 

 totx, and conversely every factor of that determinant of degree equal to 

 the order ofx is identically zero. 



3 A letter just received from M. Hermite informs me that M. Poincarre', in 

 a paper presented by him to the Institute on Monday last, takes up the 

 wondrous tale of multiple quantity so largely treated of by me in recent 

 articles in the Comptcs Rendus. The subject could not be in better hands. 

 The ball is served, and the most skilful and practised players — the Cayleys, 

 the Lipschitzes, the Poincarres, the Weyrs, the Buckheims (and who knows 

 how many more?) — stand round ready to receive it, and keep it flying in 

 the air— November 8. 



