May 2, 1889 I 



NATURE 



A Treatise on Elementary Algebra and Algebraical 

 Artifices. By Saradaranjan Ray, M.A. Two Vols. 

 (Calcutta : S. K. Lahiri and Co., 1888.) 



The work under review is also Part II. of "A Course 

 of Elementary Mathematics." Vol. I. comprises those 

 portions of elementary algebra usually to be found in 

 modern text-books up to the chapter on proportion. 

 Vol. II. includes chapters on variation, theory of equa- 

 tions, elimination, binomial theorem, and properties of 

 logarithms, as well as many others ; but those mentioned 

 will suffice to show the general scope. The author gives 

 the object he has had in view : " To create in beginners 

 a taste for algebra, and to show them the utility and 

 application of algebraical artifices." In achieving this 

 desire he has met with some degree of success. The 

 .anguage employed is simple and clear. The proofs in 

 many instances are interesting. We question the advisa- 

 bility of placing the binomial theorem and properties of 

 logarithms at the conclusion of a work which contains 

 biquadratic equations. At least an elementary chapter 

 upon the former subject should precede the theory of 

 equations, while a discussion of logarithms would be of 

 great use in its development. 



We notice one very important omission : the multi- 

 nomial theorem finds no place in these volumes. This 

 is much to be regretted, for the chapter on permutation 

 could have introduced it in an elegant and suggestive 

 manner. Again, we notice that the subject of series is 

 scarcely touched upon. One would hope in a work 

 of this scope to find a short chapter which would in- 

 clude reversion of series. However, there are many ex- 

 cellent features in the book. Chapters on " Consistency 

 and Sufficiency," and on " Identities and Equalities," are 

 novel and gratifying. The pages concerning arith- 

 metical and geometrical progressions are original and 

 inspiring ; for example, the student is taught the mean- 

 ing of the sum of n terms of a progression when n is 

 negative, and is shown that both arithmetical and geo- 

 metrical progressions possess the common property that 

 two successive terms are connected by a linear relation ; 

 from this point of view the series are then further 

 examined. 



There are many examples, as a rule well selected, with 

 occasional hints to show the learner that a little ingenuity 

 will often overcome particular difficulties. There are a 

 few occurrences of faulty printing, and some misprints 

 which do not appear in the errata. 



The students of our Eastern dependency are in pos- 

 session of a book by one of their own countrymen who 

 is a thorough master of his subject. 



LETTERS TO THE EDITOR. 



[ The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents . Neither can he undertake 

 to return, of to correspond with the writers of. rejected 

 manuscripts intended for this or any other part of NATURE. 

 No notice is taken of anonymous communications. ] 



The Sailing Flight of the Albatross. 



In Nature, vol. xxxix. p. 230, the late Mr. William Froude, 

 in a letter to Sir William Thomson, on the subject of the 

 "Soaring of Birds," gives a short account of the well-knowa 

 and often discussed sailing flight of the albatross ; and says 

 that after long consideration the only explanation which pre- 

 sented itself to his mind was, that the birds use the upward 

 current caused by the lifting of the air from the bottom of the 

 trough to the level of the crest of an advancing wave. Mr. 

 Froude by a rough calculation — the waves being supposed 10 

 feet from hollow ■ to crest — concludes that an upward current of 

 3 feet per second may be caused in this manner, and stales that 

 the bird's movements were often in accordance with this ex- 

 planation, though it was often impossible to affirm or deny the 

 accordance. 



Having had many opportur.ities of observing the flight of the 



albatross, I cannot think this explanation the true one. I have 

 often seen albatrosses sailing when the sea appeared as flat as a 

 table, with the exception of the small w aves caused by a light wind. 

 This seems fatal to Mr. Froude's explanation, which requires 

 waves of considerable size. As Mr. Froude observed, the birds 

 often sail along the crests of waves. It seemed to me that they 

 were sometimes using the current diverted upwards by the wave, 

 just as on land birds use that diverted by hedges, river-terraces, &c, 

 I will first give a description of the flight of these birds, as I 

 observed it when on board steamers running between Australia 

 and New Zealand, which are nearly always followed by numbers 

 of birds ; and then attempt an explanation. The sailing flight is 

 never to my knowledge done in a calm. I once observed the 

 effect of a gradually diminishing \\ind on their flight. The 

 steamer was going about nine knots. When the wind, which 

 was very nearly aft, became one or two knots slower than the 

 steamer, the birds, which had hitherto kept their wings perfectly 

 steady, began to flap at intervals which became shorter as the 

 wind lessened, and when it ceased they flapped almost without 

 intermission, and soon ceased to follow the vessel. 



The birds go through a series of movements which are related 

 to the direction of the wind. Starting from near the surface, 

 they rise in a slanting direction against the wind, to a height 

 which varies with the strength and direction of the wind. The 

 average seemed to me about 20 feet. Then comes immediately 

 a turn half round in a rather large circle, followed at once by a 

 rapid descent down the wind. They then take a longer or 

 shorter flight in various directions, almost touching the water. 

 After that another ascent in the same manner, and so on, 

 repeating the series of movements aJ libitum. The interval of 

 time between the ascents evidently depends on the direction of 

 the wind with regard to the course of the vessel. When the 

 wind is ahead, and the birds' velocity througii the air great, 

 being necessarily greater than the wind's velocity plus that of 

 the steamer, the interval is short. When the wind is abaft the 

 beam, and the birds' velocity much less, the interval is usually 

 much longer. As a general rule, there is a rough proportion as 

 to the favourableness of the wind and the length of time between 

 successive ascents. Also, when the wind is favourable and not 

 strong they do not rise so high as in a strong and adverse wind. 



The explanation I have to give of the movements above 

 described depends upon the well-known fact that the velocity 

 of the wind at the surface is diminished by friction, so that its 

 velocity increases with the height, the rate of increase being 

 greatest near the surface. Prof. Osborne Reynolds found by 

 experiment that the wind's velocity over a grass meadow at a 

 height of 8 feet was double that at i foot.^ Over the sea, when 

 there is enough wind to roughen the surface, the drag on the 

 lowest stratum is probably greater than over a grass meadow, 

 on account of the motion communicated to the water. This effect 

 of friction makes clear the object of the ascents against, and the 

 descents with, the wind. For, as a bird rises, he enters currents 

 of wind which increase in velocity with the height, in a direction 

 contrary to his own motion, so that the loss of velocity conse- 

 quent on rising, and which would take place in still air, is partly 

 — or perhaps, when the wind is strong, wholly — made good. The 

 bird thus gains energy of position, which is converted into 

 energy of motion by descending. A bird's ascent against the 

 wind may be compared with the ascent of a particle up an 

 incline, while the incline itself is accelerated in a horizontal 

 direction opposite to that of the particle's motion, thereby 

 enabling it to reach a height greater than that due to the initial 

 velocity. The albatross does not go on rising until his velocity 

 is nearly exhausted, but makes a half-turn at great speed pre- 

 vious to his descent. Thus the quickness of the ascent, with, 

 as Mr. Froude says, scarcely if any apparent loss of speed, is 

 explained. By making a slanting dcbcent with the wind, the 

 bird carries with him the velocity of the faster-moving wind of 

 the high level into the slower-moving wind near the surface ; 

 and thus increases his velocity through the air, to which is to be 

 added that due to his fall. Thus, if resistance is left out of 

 account, the bird's velocity since he began to ascend would have 

 been increased by twice the difference in the velocity of the wind 

 near the surface, and at the height to which he rises. And as 

 the power of overcoming resistance varies as the square of the 

 velocity, the addition of several feet per second to the bird's 

 already high velocity is equivalent to much more energy than 

 is lost in the few seconds occupied by the rise, turn, and fall ; 



' "On the Refraction of Sound by the .\tm.isphere," by Prof. Osborne 

 Reynolds. Kead before the Royal Society, Apiil aj, 1874. 



