August j, 19 13] 



NATURE 



595 



factory engine-room, at a speed increased to 195 revo- 

 lutions per minute. It developed frequently 50 brake 

 horse-power with coal gas for several hours together. 

 Sine, then the engine has been taken to Cam- 

 bridge, and is now engaged in regular service with a 

 suction-producer, driving the workshops, and produc- 

 ing electric current for the engineering laboratory. 

 It is left to itself like an ordinary gas-engine, giving 

 no trouble at all, and has been in regular work for 

 two vears, the total time of running being 5000 hours. 

 Judging from the success which has so far been 

 obtained, it seems likely that Prof. Hopkinson's 

 method of cooling the cylinder will revolutionise the 

 design and construction of large gas-engine cylinders. 



RECENT PAPERS ON VERTEBRATE^ 

 PALAEONTOLOGY. 



AVERY remarkable announcement is made by 

 Mr. J. \V. Gidley in vol. lx., No. 27, of the 

 Smithsonian Miscellaneous Collections, namely that 

 an associated series of five upper cheek-teeth of a 

 large ruminant from a Pleistocene cave-deposit near 

 Cumberland, Maryland, U.S.A., indicate an antelope 

 apparently closely related to the elands of Africa. So 

 near, indeed, is the resemblance that the author deems 

 himself justified in referring the fossil to the existing 

 genus, under the name of Taurotragus americanus ■ 

 and the plate showing these teeth alongside those of 

 the existing T. oryx goes a long way in confirming 

 his conclusion. It should have been mentioned that 

 the present writer (see Cat. Siwalik Vert. Ind. Mus., 

 part i., p. 18S5) has provisionally referred certain 

 teeth from the Indian Siwaliks to Taurotragus 

 ( = Oreas); and if the identification be correct, it would 

 explain how eland might have reached America from 

 Asia by the Bering Sea route. Mr. Gidlev quotes the 

 occurrence in the Pleistocene of Nevada of remains of 

 certain ruminants described as Ilingoceros and Spheno- 

 phalus as corroborative evidence of the former exist- 

 tence of tragelaphine antelopes in America ; but he 

 omits to mention that although these genera were 

 at first assigned to that group, they have been subse- 

 quently regarded as akin to the American family 

 Antilocaprida' (Merriam, Bull. Dept. Geol. Univ. 

 California, vol. vi., p. 292). If this be correct, is it 

 quite impossible that the supposed eland represents 

 another member of the same group? 



In a second communication (op cit.. No. 26) Mr. 

 Gidley records the occurrence of a toe-bone of a camel 

 in a superficial deposit at the mouth of Old Crow 

 River, in the Yukon Territory, in association with 

 remains of mammoth, horse, and bison. The occur- 

 rence of the camel-bone confirms "the theory of the 

 existence of a wide Asiatic-Alaskan land connection of 

 comparatively recent date, which for a very consider- 

 able length of time served as a great highway for the 

 free transmission of mammals between America and 

 the Old World." 



As being only in part palaontological, brief notice 

 must suffice for a paper, by Mr. K. S. Bardenfleth, on 

 the form of the carnassial tooth in Carnivora, pub- 

 lished in Vidensk. Meddel. Dansk. naturh. Foren, 

 vol. Ixv., pp. 67-1 1 1. After reviewing the various 

 theories of the homology of tooth-cusps, the author 

 proceeds to observe that in order to demonstrate that 

 the simple reptilian tooth-cone is represented by the 

 middle one of the three longitudinally arranged cusps 

 of the Purbeck Triconodon, and that the tritubercular 

 crown has been formed by rotation of the other two, 

 indisputable evidence has yet to be furnished, "first, of 

 the Triconodon-like forms being the ancestors of 

 Dryolestes, &c. ; second, of the supposed protocone 

 and protoconid of these being really homologous with 

 NO. 2284, VOL. qi] 



the median cusp of Triconodon. One can scarcely 

 imagine how such a rotation could take place, and 

 if Gidley is right in his interpretation of the molar 

 cusps of Dryolestes, the rotation has not taken place, 

 but the so-called protocone is a secondary acquirement ; 

 the true protocone is still to be sought in the central 

 one of the three outer cusps. If this holds good the 

 whole nomenclature and theory of Osborn falls to 

 the ground ; neither protocone nor protoconid are then 

 identical with the reptilian cone." 



Three papers, by Dr. R. Broom, form part 6 of 

 vol. vii. of the Annals of the South African Museum, 

 and relate to the extinct reptiles of the same country. 

 In the first of the triad the author shows that while 

 in Pariasaurus the digital formula is 2.3.3.4.3, in the 

 allied Propappus it is probably 2.3.4.5.3. I n tne second 

 he describes, as Noteosaurus africanus, a new genus 

 allied to Mesosaurus, of which three of the known 

 species are South African, while the fourth is 

 Brazilian. The last paper comprises a classified list 

 of the early Mesozoic reptiles of South Africa, which, 

 apart from dinosaurs, crocodiles, rhynchocephalians, 

 &c, are arranged in no fewer than nine ordinal 

 groups, brigaded in three " superorders." R. L. 



AN ALGEBRA FOR PHYSICISTS. 1 



THE principal novelties in Dr. Macfarlane's calculus 

 are that a distinction is made between linear and 

 cyclic successions of vectors, and that the commutative 

 law of addition, as well as that of multiplication, is 

 abandoned. To express what most vectorists write 

 B + a = a + 0, Dr. Macfarlane writes 2(3 + <i) = 2(« + <3). 

 Thus a 4-/3 -a is not the same as 0, but either three 

 sides of a parallelogram, or three coinitial vectors, 

 according as we take linear or cyclical succession. 

 By introducing some subsidiary and rather artificial 

 rules, the author is able to get formulae that are, in 

 appearance, analogous to the binomial and exponential 

 theorems, and so on. 



The actual divergence from quaternion results is not 

 verygreat, as may be easily shown by an example. !.■ 1 

 \ be a scalar, <* a unit vector, and let exp(.va) be defined 

 tomeanXH»/n!. Then exp(.va) = cos x+a sin ,r. and 

 if y is another scalar, exp(xa).exp(ya) = exp(yn 

 exp(xa) = exp { (,v + y)a } = cos (x + y) + a sin {x + y). 



But, if is another unit vector, 



exp(.ra) exp(_y3) = cos x cos y 4- a sin x cos_v + 

 3 cosa' sin_y +n3 sin x sin^, 

 which differs from exp(y/3).exp(.va), while both, in 

 general, differ from exp(.va 4- y/3) : the latter, observe, 

 bring by definition the same as exp(y0 4- *<"). Dr. 

 Macfarlane, after writing down his exponential. 

 formula, breaks it up into four parts, practically the 

 same as the four given by the quaternion formula 

 above, when writ ten in the form — 



exp(.rn) exp(_)'3) = (cos x cost' + sin x sin y Sn3) + 

 a sin i-cos_>' + 3 cos x sin _r + Vnftsin x sin y. 



It must be left to physicists themselves to decide 

 whether Dr. Macfarlane's new algebra is superior to 

 those already available ; the need of a sign to express 

 a resultant is a rather severe handicap. To the pure 

 analvst it presents the appearance of a con- 

 glomerate, though possibly, with a change of nota- 

 tion, it could be fitted into a place in the family of 

 linear associative algebras. One thing ought to be 

 said: it is not, properly speaking, an "extension" 

 of quaternions. Analytically, the calculus of_ quater- 

 nions is a linear algebra of a perfectly definite type, 



' (1) " Account of Researches in the Algebra of Physics," I.-III. (Repiint 

 from lourn. Wash. Ac. »f Pc, 1012 ) 



(5) "On Vector-analysis ss Generalised Algebra" (Intern. Congress of 

 Mathematicians, 1912.) By Dr. A. Macfarlane. 



