Jan. 3, 1889] 



NATURE 



219 



the same and subsequent volumes various memoirs on 

 binary and ternary quantics, including papers (by him- 

 self, with the aid of P'ranklin) containing tables of the 

 numerical generating functions for binary quantics of the 

 first ten orders, and for simultaneous binary quantics of 

 , the first four orders, &c. The memoir (vols. ii. and iii.) 

 fl on "Ternary cubic-form equations" is connected with 

 r some early papers relating to the theory of numbers. 

 We have in it the theory of residuation on a cubic curve, 

 and the beautiful chain-rule of rational derivation ; viz. from 

 an arbitrary point i on the curve it is possible to derive 

 the singly infinite series of points (i, 2, 4, 5, ... 3/ ± i) 

 h that the chord through any two points, /«, «, again 

 ets the curve in a point in 4- », m ~ n (whichever 

 number is not divisible by 3) of the series ; moreover, 

 the coordinates of any point w are rational and integral 

 functions of the degree m- of those of the point i. 



There is in vol. v. the memoir, " A Constructive Theory 



1; f>t" Partitions arranged in three acts, an Interact in two 



I^H|tts, and an Exodion," and in vol. vi. we have" Lectures 



I^Hthe Principles of Universal Algebra" (referring to a 



I^Hkrse of lectures on multinomial quantity, in the year 



I^Bi)- The memoir is incomplete, but the general 



f ^ffeories of nullity and vacuity, and of the corpus formed 



by two independent matrices of the same order, are 



sketched out ; and there are in the Comptes rendus of the 



French Academy later papers containing developments 



«f various points of the theory, — the conception of 



i^fcivellators " may be referred to. 



IHRrhe last-mentioned paper in the American Mathema- 

 \Wical Journal was published subsequently to Sylvester's 

 return to England on his appointment as Savilian Pro- 

 fessor of Mathematics at Oxford. In December 1886, 

 he gave there a public lecture containing an outline of 

 his new theory of reciprocants (reported in Nature, 

 January 7, 1887), and the lectures since delivered are 

 pubhshed under the title, " Lectures on the Theory of 

 Reciprocants" (reported by J. Hammond), same Journal 

 vols. viii. to X.; thirty-three .lectures actually delivered, 

 entire or in abstract, in the course of three terms, to a 

 class in the University, with a concluding so-called lec- 

 ture 34, which is due to Hammond. The subject, as 

 is well known, is that of the functions of a dependent 

 variable, J, and its differential coefficients, /,/', . . ., 

 ill regard to x (or, rather, the functions of /,/', . . . ), 

 which remain unaltered by the interchange of the 

 variables :r and J : this is a less stringent condition than 

 Uiat imposed by Halphen (" Th^se," 1878) on his differ- 

 ential invariants, and the theory is accordingly a more 

 extensive one. A passage may be quoted : — " One is 

 surprised to reflect on the change which is come over 

 Algebra in the last quarter of a century. It is now pos- 

 sible to enlarge to an almost unlimited extent on any 

 branch of it. These thirty lectures, embracing only a 

 feagment of the theory of reciprocants, might be com- 

 pared to an unfinished epic in thirty cantos. Does it not 

 seem as if Algebra had attained to the dignity of a fine 

 art, in which the workman has a free hand to develop his 

 conceptions, as in a musical theme or a subject for 

 painting 'i Formerly, it consisted in detached theorems, 

 but nowadays it has reached a point in which every 

 properly-developed algebraical composition, like a skilful 

 landscape, is expected to suggest the notion of an infinite 



distance lying beyond the limits of the canvas." And, 

 indeed, the theory has already spread itself out far and 

 wide, not only in these lectures by its founder, but in 

 various papers by auditors of them, and others, — Elliott, 

 Hammond, Leudesdorf, Rogers, Macmahon, Berry, 

 Forsyth. 



Sylvester's latest important investigations relate to 

 the Hamiltonian numbers : there is a memoir, Crelle^ 

 t. c. (1887), and, by Sylvester and Hammond jointly, 

 two memoirs in the Philosophical Transactions. The 

 subject is that of the series of numbers 2, 3, 5, n, 47, 

 923, calculated thus far by Sir W. R. Hamilton in his 

 well-known Report to the British Association, on Jerrard's 

 method. A formula for the independent calculation of 

 any term of the series was obtained by Sylvester, but the 

 remarkable law by means of a generating function was 

 discovered by Hammond, viz. E^,, Ej, E,, . . . , being 

 the series 3, 4, 6, . . . of the foregoing numbers, eack 

 increased by unity ; then these are calculated by the 

 formula (i -tf" ■\- /(i - tf^-\-t\\ - if' +...== 1-21, 

 equating the powers of I on the two sides respectively : 

 observe the paradox, I = ^, then the formula gives o = 

 sum of a series of positive powers of i. 



Enough has been said to call to mind some of Sylves- 

 ter's achievements in mathematical science. Nothing fur- 

 ther has been attempted in the foregoing very imperfect 

 sketch. A. Cayley, 



THE CREMATION OF THE DEAD. 



The Cremation of the Dead. By Hugo Erichsen, M.D. 

 (Detroit: D. O. Haynes and Co., 1887.) 



THIS book is an appeal to the general public on the 

 propriety of introducing the practice of cremation, 

 universally, into civilized communities ; or, as the author 

 puts it, " it is a plea for the burning of the dead." He 

 considers, and we are inclined to think he is right, that 

 the period of fanatic and fierce opposition to crematioa 

 has passed, and has made way for a calm consideratioa 

 of the subject. In 1874, he tells us, a Persian gentleman 

 then resident in one of the Eastern States of the free 

 and great Republic of America, who wanted to have 

 his wife cremated, was compelled by an ignorant mob to 

 resort to interment ; but now the feeling has changed- 



In our own country the same sensible desire to discuss 

 the question of cremation, fully and freely, is fairly estab- 

 lished at the present time ; and so greatly has prejudice 

 disappeared, that now the act of cremation has been 

 carried out over fifty times at the Woking Cemetery- 

 alone. As Sir Spencer Wells shows, in an introductory 

 chapter which he has written for the work before us, the 

 obstacle of law in England against cremation has been 

 removed, and relatives may resort to the cremation of 

 their dead without any unreasonable impediments. 



In saying so much in favour of freedom in regard to 

 cremation, we must, however, in this country confine the 

 freedom to the voice of the living. The wishes of the 

 dead, though they may have been delivered up to the last 

 moment in favour of cremation, and may even have been 

 ordered in the will of the deceased, have no legal weight 

 with the survivors. The writer of this article was called, 

 quite recently, to see a lady who had rather suddenly died, 



