February 9, 1893] 



NATURE 



339 



-Ci:t2 + Cr-Cj = o, if A = - 27C32 + iSCiQCg- 

 467C3-4C/i C,-V,V, .? = 2Ci3-9C,C2 + 27C3, ^=9^3 

 — 3C1C2, the relations in question are 



l\^ =. - 27 A, v.? = i{S + K3), Vi> = as - V,) ; 



f^l = ^1 + W^X, + 0)4-3, ^2 = ^'1 + ®^2 + (O^^if 



V,= T+ Z'cix^Kv, + x.^x, + ;t32^2) + 3«'(^i''-t-2 + X.?X; 



+ x^^x,); 

 r, = i\c^ +v^+ f;;, x, = J{Ci 4- CO Fj + 0,2 K,!, 



IJy means of this theorem and certain elementary prin- 

 ciples of the theory of substitutions an elegant and simple 

 demonstration can be given of Abel's theorem that the 

 solution by radicals of the general equation of the «'h 

 degree is impossible when // > 4 : see § 217 of the work 

 before us. 



Although the theory of substitutions bears, as we have 

 just shown, on some of the oldest and most interesting of 

 the problems of algebra, it has been comparatively little 

 studied, especially by English speaking mathematicians. 

 Dr. Cole has therefore rendered us a service of great 

 importance by translating one of the standard treatises 

 on this subject. Of the three that were at his disposal 

 we think that he has chosen the one most likely to be 

 useful to a beginner. While Serret in his " Higher 

 Algebra" and Jordan in his "Traitd" treat the theory 

 from an abstract and more general point of view. Dr. 

 Netto constantly associates with the substitution the 

 function on which it is supposed to operate. This gives 

 a powerful concrete aid to the comprehension of the pro- 

 positions of the abstract theory and also helps the student 

 to grasp their application. The great danger in subjects 

 of such generality is that the stream of theorems is apt to 

 run off the mind of the learner without soaking in, like 

 water ofif the proverbial duck's back. 



Dr. Netto's book will be found to contain all the ordi- 

 nary theorems regarding the classification of substitutions, 

 e.g. the existence of groups, transitive and intransitive, 

 primitive and non-primitive, simple and compound; the 

 theory of the algebraic relations between the values of 

 multiple-valued functions and between functions belong- 

 ing to or included in the same family ; and also a con- 

 siderable number of theorems regarding special groups. 

 The applications embrace the theory of resolvents in 

 general and of the Galois resolvent in particular; the 

 general theory of the solvability of equations by means of 

 radicals ; the theory of the group of an equation and a 

 discussion of the criteria of solvability ; besides special 

 applications to the cyclotomic and Abelian equations, and 

 to equations three roots of which are connected by a 

 rational relation. 



The translation has been admirably done, both from 

 the linguistic and from the mathematical point of view. 

 We found, it is true, here and there passages which were 

 somewhat obscure ; biit in every case, on comparing with 

 the original, we found the rendering to be absolutely 

 faithful. Such obscurities therefore must be charged 

 either to the author, or to the nature of the subject, or to 

 the idiosyncrasy of the critic, and not to the translator. 

 We congratulate Mr. Cole on the successful completion 

 of his arduous task, and heartily recommend the result 

 to every lover of the most ancient and the most beautiful 

 of all the sciences. G. Ch. 



NO. I 2 15, VOL. 47] 



THE BRAIN IN MUDFISHES. 

 Das Centralnervensystem von Protopterus annectens ; 

 eine vergleichend Anatomische Studie. Von Dr. Rudolf 

 Burckhardt. (Berlin: R. Friedlander und Sohn, 1892.) 



THE Mudfishes, Dipnoi, from many peculiarities in 

 their structure, have attracted the especial attention 

 of anatomists and zoologists. Important monographs 

 on Lepidosiren have been written by Owen and Wieders- 

 heim, whilst Huxley, Gunther, and Beauregard have 

 described the anatomy of Ceratodus. Serres, in 1863, 

 made a contribution to the anatomy of the nervous 

 system of Protopterus, Fulliquet in 1886, and Parker in 

 1888, have also added to our knowledge of its struc- 

 ture ; and now Dr. Burckhardt has published a well- 

 illustrated monograph on the central nervous system of 

 Protopterus annectens. He had obtained an ample 

 supply of this fish from Herr W. Jezler, a merchant 

 whose business engagements had taken him to the 

 neighbourhood of Bathurst, Senegambia. On more than 

 one occasion Dr. Burckhardt had received living fish, so 

 that he was able to study the microscopic anatomy by 

 the use of the most recent technical methods, and has 

 thus added materially to our knowledge of the brain of 

 this animal. 



The author found, in the anterior horn of grey matter 

 of the spinal cord, remarkably large nerve-cells, which 

 possessed both branching protoplasm processes and an 

 axial-cylinder process. In the lateral and posterior horns 

 nerve-cells somewhat smaller in size were seen. The 

 medulla oblongata gave origin to nerves which he names 

 hypoglossal, vagus, glosso-pharyngeal, acusticofacialis, 

 and trigeminus. He also describes two slender nerves as 

 abducens and trochlearis, so that the Dipnoi are not, as 

 some have said, destitute of these nerves. The cere- 

 bellum formed the anterior boundary of the 4th ventricle. 

 Large nerve-cells, corresponding to those of Purkinje in 

 the mammalian brain, were not seen. The mid-brain was 

 distinct, and gave origin to a root of the trigeminus, to 

 the optic tract and to the oculo-motor nerve : grey matter 

 containing nerve-cells was grouped around the aqueduct 

 of Sylvius. 



Whilst Protopterus corresponded closely with the 

 lowest vetebrates in the regions of the mid and hind 

 brains it presented striking peculiarities in the pineal 

 region. The roof of the 3rd ventricle was complicated, 

 and possessed a velum, which represented a middle 

 choroid plexus ; a conarium, and a structure like that 

 which Edinger has named " Zirbelpolster." The epiphysis 

 {Zirbel) was attached to the skull by the arachnoid 

 membrane. 



The fore brain was well developed, and divided into 

 two hemispheres. He recognized in it a posterior ventral 

 swelling, which, because it contained cells similar to 

 those found in the dentate gyrus {fascia dentatd) of the 

 higher brains, he describes as a lobus hippocampi. He 

 distinguished a fissure which separated the lobus olfac- 

 torius from the pallial part of the hemisphere, so that he 

 harmonizes the fore brain in its fundamental divisions 

 with the mammalian brain as described by Broca and 

 Turner. He directs attention to an elevation ventrad 

 of the lobus olfactorius, which he calls the lobus post- 

 olfactorius. This lobe is also found in the brains of 



