Oct, 13, 1887] 



NATURE 



557 



yellow, orange, red. A little later, the same night, on the brow 

 of the ridge, in the faint mist which rose in masses, the bow was 

 again seen vividly, against a background of trees ; the bow being 

 within 40 paces of the observer. W. G. Brown. 



Washington and Lee University, Lexington, Va. 

 September 23. 



Destruction of Young Fish. 



May I call your attention to the wholesale destruction of 

 young fish which is carried on to a great extent round our coast ? 

 A few facts may not be out of place. Recently I have been visit- 

 ing a small fishing village on the east coast, and have carefully 

 noted the amount of young fish rejected by the fishermen on their 

 return from trawling and shrimping. For example, from a,\ pecks 

 of shrimps no less than 793 flat-fish (dabs, soles, and turbots) 

 were thrown on the beach useless ; to this must be added about 

 200 whiting and an amount of young cod, herring, and skate 

 beyond my power to count. Surely something can be done to 

 remedy this ! I: is well known to the fishermen that the net 

 does not injure the fish ; so tiiat before landing, if the net was 

 roughly examined, all young fish could be thrown into the water 

 again. David Wilson-Barker. 



66 Gloucester Crescent, Regent's Park, N.W. 



ON HAMILTON'S NUMBERS. 



■pOLLOWING in the footsteps of Hamilton in his 

 ^ Report to the British Association, contained in the 

 Proceedings for the year 1836, we may arrive at a solu- 

 tion, in a certain sense the simplest, of a problem in 

 algebra the origin of which reaches back to Tschirn- 

 hausen, bom 1651, deceased 1708. Every tyro knows 

 how a quadratic equation, and all equations of a superior 

 degree thereto, may be transformed into another in which 

 the second term is wanting. Tschirnhausen showed that 

 a cubic equation, and all equations superior in degree to 

 the cubic, might be deprived of their second and third 

 terms by solving linear and quadratic equations. Then 

 over a century later Bring, of the University of Lund, in 

 1786 showed that every equation of the 5th, or any higher 

 degree, might be deprived of its first three terms by means 

 of solving certain cubic, quadratic, and linear equations.^ 

 What, then, it may be asked, is the law of the progression 

 of which the three first terms are 2, 3, 5 ? What is the 

 lowest degree an equation can have in order that it may 

 admit of being deprived of four consecutive terms by aid 

 of equations of the ist, 2nd, 3rd, and 4th degrees, or more 

 generally of i consecutive terms by aid of equations of 

 the 1st, 2nd, 3rd, . . . and /th degrees, i.e. by equations 

 none of a higher degree than the zth ? '■' 



' In a letter to Leibnitz (1677), which I have not seen, and the Acta 

 Eruditorunt for 1683. 



* How the elevation of the degree of the equation to be transformed 

 makes it possible to abolish a greater number (m) of terms by an auxiliary 

 system of equations of degrees none eicceeding /* will be understood if we 

 consider the cases of a quintic and quartic. 



Supposing {x, l)° to be a given quintic, on writing 



we obtain, by elimination of X^ (o, )8, "y, 8, if = O, and any solution of 

 his equation will enable us, by a well-known pr jcess, to find jp by a linear 

 equation. 



If we select any letter, a, of the five we may equate it to a linear function of 

 y, ;8, % 5, *, so as to obtain 



r' + C/S. 7. 5, f^^ + C/S. 7. 8, Oy + (5, 7, 8, «)V + (J3, 7. 8, «)' = o. 



If in this equation we can find any system of ratios /3 : y : 8 : « such that 

 (/8, 7, 5, if — O, and (3, 7, 8, e)' = o, we can find y by solving a 

 trinomial quintic, and therefore a system of admissible ratios o : p : 7 : : « 

 becomes known. 



All that is requisite therefore is to be able to obtain any point whatever of 

 intersection of two given quadratic and cubic surfaces represented by 

 (3, 7, 8, 6)- and (/3, 7, 5, e)' which obviously may be done by first finding 

 a point (any point) in the quadratic surface (which only necessitates S)lving 

 some quadratic equation or other) ; second, at this point drawing a right line 

 (either one of a pair) lying on the surface, which may be effected by a well- 

 known method involving only the solution of a quadratic ; and third, finding 

 any one of the three intersections of such line with the cubic surface. 



Thus, then, by solving quadratic and cubic equations a quintic may be 



In the 1 00th volume of Crelle's Journal {\Z%6) I have 

 shown that the progression continued as far as the*case 

 of eight terms being abolished is as follows — 



2, 3, 5, 10, 44, 905, 409181, 83762797734. 



These, with the exception of the three first, are not 

 exactly what I call Hamilton's numbers, but serve to lead 

 up to them. 



Hamilton's numbers are — 



2, 3, 5, II, 47, 923, 409619, 83763206255, . . . 



I will endeavour to explain wherein the difference 

 consists between the two series. 



Whilst it is true that four terms may be abolished in an 

 equation of the loth degree without solving equations 

 beyond the 4th degree, there is this difference in favour 

 of equations of the nth or any higher degree, viz. that 

 fewer biquadratics will be required for them than in the 

 case of an equation limited to the loth degree. And so 

 in general whether we take, as our inferior limit to the 

 degree of the equation to be transformed, the zth number 

 in the upper series or the iih. number in the lower one — 

 whilst in neither case it will be necessary to solve any 

 equations of a degree exceeding z — the total system in the 

 latter case will be of a simpler character than in the 

 former. 



The numbers which I have named in honour of 

 Hamilton rhay be obtained by a process exhibited in the 

 table below — 



We may now isolate the greatest figure which occurs in 

 each column, and in this may we obtain the numbers 

 I, I, 2, 6, 36 . . . which I call hypotenusal numbers; 

 then adding these numbers together and increasing each 

 sum so obtained by unity we arrive at the so-called 

 Hamilton's numbers, viz. 2, 3, 5, 11,47. .. . Now the 

 question arises as to how they may be calculated ; for 

 obviously the crude method above given will be impos- 

 sible to carry out in practice beyond the first few numbers 

 in the scale. The method of generating functions — of 

 which the idea occurred first to my coadjutor Mr. James 

 Hammond, which certainly ought not to, and probably in 

 the long run could not, have escaped me — leads to a 

 wonderfully beautiful law, by means of which these 

 numbers may be derived successively each from those 

 that go before, just as is the case with Bernouilli's 

 numbers. 



The simplest and best mode of proceeding is as 



deprived of three consecutive terms. But not so a quartic ; for in the case 

 of a quartic we could not (with any real advantage) use a subsidiary equation 

 ofa higher degree than the 3rd. we should thus have only three letters, 3, 7» 8, 

 instead of four in the equation in ^, and to make (i9, 7, 8)* = 0, (3,7,8)' = 0» 

 simultaneously, is the problem of finding an intersection of a quadratic and 

 a cubic curve, which necessitates the solution of an equation of the 6th 

 degree. 



In the case of the quintic it may be well to notice that the ratios 

 o : i3 : 7 : 6 : € will not be all real, and consequently the trinomial quintic 

 into which the original one has been transformed will not have its coefficients 

 real, unless the quadric surface is a hyperboloid of one sheet (since it is 

 only that species of quadric surfaces which contains real straight lines) ; 

 and I have shown in my paper in Crelle that this is the case then, and 

 then only, when the original quintic has four imaginary roots. 



» If the number of equations of degrees i, ; — i, i — 2 to be solved 



in the one case reckoned in descending order are /t, i, c, .... i, ... . and 

 in the other a', b\ c', . . . . I . . . . respectively, if /, I are the two first 

 corresponding numbers which are not identical /' will be less than /. 



