yan. 11,1872] 



NATURE 



199 



•nedusa also. Then the budding from the proboscis of 

 Dysmorphosa fiil^iiyaiis A. Ag-. is noticed, and the 

 nomadic or free-floating hydroid Nanomia also. Synapta, 

 amongst the Holothurians, is noticed on account of its 

 curious sand-ring clothing. "They live in very coarse 

 mud, but they surround themselves with a thin envelope 

 of fine sand, which they form by selecting the smaller 

 particles with their tentacles, and making a ring around 

 their anterior extremity. This ring they then push down 

 along the length of the body, and continue the process, 

 adding ring after ring, till they have entirely encircled 

 themselves with a sand tube. They move the rings down 

 partly by means of contractions of the body, but also by 

 the aid of innumerable appendages over the whole surface. 

 To the naked eye these appendages appear like little 

 specks on the skin ; but under the microscope they are 

 seen to be little warts projecting from the surface, each 

 one containing a little anchor with the arms turned up- 

 ward. Around the mouth the warts are larger, but do 

 not contain any anchors." " By means of these appen- 

 dages, though aided also by the contractions of the body, 

 the Synaptx move through the mud, and collect around 

 themselves the sand tube in which they are encased." 

 They gorge themselves with mud and sand for the sake 

 of the nutritious substances they may contain. The office 

 of the pedicellaria: of the Sea Urchin is well described, as 

 follows : — " If we watch the Sea Urchin after he has been 

 feeding, we shall learn at least one of the offices which 

 this singular organ performs in the general economy of 

 the animal. That part of his food which he ejects passes 

 out at an opening on the summit of the body, in the small 

 area where all the zones converge. The rejected particle 

 is received on one of these little forks, which closes upon 

 it like a forceps, and it is passed on from one to the other 

 down the side of the body till it is dropped off into the 

 water. Nothing is more curious and entertaining than to 

 watch the neatness and accuracy with which this process 

 is performed. One may see the rejected bits of food 

 passing rapidly along the lines upon which these pedi- 

 cellaria; occur in greatest number, as if they were so many 

 little roads for the carrying away of the refuse matters ; 

 nor do the forks cease from their labour till the surface of 

 the animal is completely clean and free from any foreign 

 substance." Some higher animals might take a profitable 

 lesson from the Urchin. The Crinoids are passed by 

 rather briefly. The existence of Comatulas from Green- 

 land to South Carolina is mentioned, but the authors do 

 not appear to have devoted special attention to them. A 

 very excellent notice of the embryology of the Echino- 

 dermata precedes the last chapter, which consists of a brief 

 resume of the distribution of life in the ocean. The book 

 might be taken as a model by many European naturalists 

 who write popular works, for there is a vast amount of 

 philosophy in it. The authors have not contented them- 

 selves with serving up a number of "wonders "for the 

 public bewilderment ; nor have they simply given us a 

 series of descriptions of forms, as is the practice especially 

 amongst those who trade upon butterflies and beetles ; 

 but they have taken a vast amount of trouble in explain- 

 ing the development and embryology of the Invertebrata 

 which have come under their notice. In fact, they have 

 given a reasonable amount of bread with their " sack." 



P, M. D. 



EARNSHAW'S DIFFERENTIAL EQUATIONS 



Partial Differential Equations. An Essay towards an 

 entirely new Method of Integrating them. By S. 

 Earnshaw, M.A. (Macmillan and Co., 1871.) 

 "T^HE present work, as its title indicates, contains a 

 -»- detailed explanation of a new method of integrating 

 Partial Differential Equations ; it is in no sense a text- 

 book or introduction to the subject. The author's object 

 is not to collect and describe the known methods, but to 

 develop a new one. The principle of the method is easily 

 explained and understood. The independent variables 

 in the given differential equation being t, x, y,z . . ., we 

 can transform it so that the new independent variables 

 are /, ^, V, f, . . ., by equations of the form D^u = d^u 

 -f d^u . D^x -f . . . ; but the practical application of 

 the method consists in comparing the original equation 

 with the equation last written, and thus determining rela- 

 tions from which, by the elimination of ^, i;, f . . ., the in- 

 tegral of the original differential equation is found. The 

 quantities /, |, »;, f, . . ., with the exception of the one with 

 regard to which the differentiation is being performed, 

 are treated as constants, and are here called quasi-constants 

 (semi-constants we should have preferred). Mr. Earnshaw, 

 as is apparent from the equation of transformation quoted 

 above, adopts rfwhen the differentiation is with regard to 

 the old variables, and D when with regard to the new ; 

 the suffix notation for differential coefficients is also made 

 use of For this latter departure from custom the author 

 in the preface offers an apology, and states that he has 

 been warned that it " will form a serious hindrance to the 

 acceptableness of the present work." This fear we think 

 is groundless ; the notation is not inconvenient in such in- 

 vestigations as the present, as it somewhat simplifies the 

 appearance of the equations without rendering the ana- 

 lysis more difficult to follow. 



In the first few chapters the method is applied to the 

 integration of numerous equations of the first and second 

 orders, and throughout the book the applications to par- 

 ticular cases are so numerous that whole chapters con- 

 sist entirely of " examples worked out." This excessive 

 number of examples is a drawback, as many of them (for 

 instance, all in Chapter V., which treats of linear equa- 

 tions of the second order with constant coefficients) can 

 be more simply and perfectly discussed by Boole's symbolic 

 and other methods. The reader is also left in doubt 

 as to how far the examples have been chosen so as to 

 suit the method of solution here adopted. In the deve- 

 lopment of a new principle it is always a matter of great 

 importance to point out the cases in which it enables us 

 to obtain results previously beyond our reach, and also the 

 cases in which the previous methods are preferable. This 

 Mr. Earnshaw does not appear to have done ; he has 

 integrated a great number of equations, many of which, 

 however, are capable of solution by well-known methods 

 in as straightforward a way as ordinary quadratics in 

 algebra. It is, for such reasons as these, generally desirable 

 that original mathematical investigations should appear 

 first in the memoirs of a Society or other recognised oro-an 

 where the new matter is distinctly stated, rather than in the 

 form of a book where there is nothing to check the temp- 

 tation to overburden the explanation with examples. Mr. 

 Earnshaw claims to have for the first time integrated in 



