262- 



NATURE 



\jfan. 1 6, 1890 



It will be noticed that in this division the quantity /"(v) has qlisappeared. 

 Now, from above, 



/(<^) = /<^(sec<^)« + i, 

 and therefore 



-/>)= /^" +(« + i)tan<|,, 



and 



Hence 



n-yl = l(^\ +in + I) tan 7. 



rF{<}>)d<l> ~ f^ ip^- "-q^- ") = (sec 7)"' 1 1 + ^\ (a - )8)2|_ 2/w ( «^^)^ tan 7 + 2w ;7+'i (tan 7)2 + ;«(;«TT(sec 7)^ - w ) ]} 



= (sec7)"' j I + •5V(o - )3)2J 2lm(--~\ tan 7 + ;«(;« + 2;/ + 3)(sec 7)- - m{m +2« + 2) l- 



Now make m + n + i = 2, 



or w = - (« - i), and we have 



r «^(sec <?))'•' ^ -^-^^ (/-" - /- ") = (cos 7)« -1 j I - ^\(a - ^f[2l{ti - i)(^~\ tan 7 + (« - i)(« + 4)(sec yf - (« - i)(« + 3)1 1 • 



In this make 1=2, and I = I, successively, and we obtain the 

 same expressions for X and T as before. 



The case thus treated is not one of mere curiosity, but is 

 practically important. From theoretical considerations, Newton 

 concluded that the resistance of the air to the motion of pro- 

 jectiles is proportional to the square of the velocity, and very 

 little progress has been made in the theory of the subject since 

 his time. Experiments have shown that the relation between 

 the velocity of a projectile and the resistance offered by the air 

 to its motion is far from being so simple as that given by the 

 theory. The most extensive and accurate series of such experi- 

 ments which we have are those made by Mr. Bashforth by 

 means of his chronograph, which measures with the greatest 

 precision the times taken by the same projectile in passing 

 over several successive arcs in the course of its flight. In 

 a summary of his results for ogival-headed shot, struck with a 

 radius of i J diameters, given in Nature (vol. xxxiii. pp. 605, 

 606), Mr. Bashforth concludes that the resistance may be ap- 

 proximately represented by supposing it to vary as one power of 

 the velocity when that velocity lies between certain limits, as 

 another power when the velocity lies between certain other 

 limits, and so on. 



Thus, if V denote the velocity expressed in feet per second, 

 d the diameter of the shot in inches, 

 and w its weight in pounds, 



and if — = c, 



w 



then, when v lies between 430 f.s. and 850 f.s., 



the resistance is nearly = 6i"kc ( 1 : 



when V lies between 850 f.s. and 1040 f.s., 



the resistance is nearly = 74*4 c ( | : 



\ 1000/ 



when V lies between 1040 f.s. and iioo f.s., 



the resistance is tiearly = 79*2 ^r ( —^JTTT" 



Viooo/ 



when V lies between iioo f.s. and 1300 f.s., 



the resistance is nearly = 108 "8 c ( ^ \ ; 

 . — — Viooo/ 



and lastly, when v lies between 1300 f.s. and 2700 f.s., 



the resistance is nearly = 141 'i; c ( ^—\ 

 ^ ^ Vicoo/ ■ 



Hence the resistance varies nearly as the square of the velocity 

 both when the velocity is less than 850 f.s., and when it is 

 greater than 1300 f.s., but the coefficient increases from 61 "3 

 in the former case, to 141 "5 in the latter. Also, the re- 

 sistance varies nearly as the cube of the velocity, both when v 

 lies between 850 f.s. and 1040 f.s., and also when it lies between 

 zioo f.s. and 1300 f.s., but the coefficient increases from 74 "4 in 



the former to 108 '8 in the latter case. Again, for velocities 

 which are nearly equal to that of sound in air, the proportionate 

 increase of the resistance is much greater than that of the 

 vehjcity. 



Mr. Bashforth remarks that the points of transition from one 

 law of resistance to another, as stated above, are somewhat 

 arbitrary, but that, if they were changed a little in either direc- 

 tion, the practical error would not be large. 



Of course, if we had at our disposal much more numerous 

 and still more accurate observations, it would be possible to 

 represent the experimental results with any degree of exactness 

 that might be desired, by subdividing the observations in'o a 

 larger number of groups, so that the limiting velocities in any 

 one group should be closer together, and that the change of the 

 index of the power of the velocity in passing from one group to 

 the next should be less aj^rupj^^ J. C. Adams. 



SOCIETIES AND ACADEMIES. 



London. 



Chemical Society, December 19, 1889. — Dr. W. J. Russell, 

 F.R. S., in the chair. — The following papers were read : — Fran- 

 gulin, by Prof. T. E. Thorpe, F.R.S., and Mr. H. H. Robinson. 

 The authors prepared the glucoside frangulin from the bark of 

 the alder buckthorn {Rha7nmis frangida), and find its formula to 

 be C22H22O9. On hydrolysis it yields a yellow product, CigHjoOs, 

 which agrees in its properties with emodin, and a sugar which 

 has the power of reducing Fehling's solution, and is not identical 

 with dextrose. — Arabinon, the saccharon of arabinose, by Mr. 

 C. O' Sullivan, F. R. S. The substance having an optical activity 

 "well above [«]/= 140," obtained by the author by the 

 hydrolysis of arable acid, and described under the name of a- 

 arabinose (Chem. Soc. Trans., 1884, 55), yields arabinose on 

 hydrolysis, and appears to bear to this carbohydrate a relation 

 similar to that which saccharon (cane sugar) bears to dextrose : 

 the author therefore terms it arabinon. It has the formula 

 CjoHjgOg, and on hydrolysis gives a yield of arabinose agreeing 

 very closely with that required by the equation CjoHjgOg + HjO 

 = aCoHjoOg. As yet it has not been obtained in a crystalline 

 state ; it has a specific rotatory power of [a]i, = 198° "8, and 100 

 parts have the same cupric reducing power as 58*8 parts of 

 dextrose. — On the identity of cerebrose and galactose, by Mr. 

 H. T. Brown, F.R.S., and Dr. G. H. Morris. The authors 

 give the results of an examination of a specimen of cerebrose, 

 prepared from phrenosin, which was placed in their hands early 

 in 1888 by Dr. Thudichum, who first isolated and crystallized 

 this substance. They show that its specific rotatory power, 

 cupric reducing power, and molecular weight as determined by 

 Raoult's method, are identical with those of galactose, thus con- 

 firming the recent work of Thierfelder, Zeit. Physiol. Chem., 14, 

 209) who has proved the sugar produced by the action of acid on 

 cerebrin to be identical with galactose. In the discussion which 

 followed the reading of the paper, Dr. Thudichum said that 

 phrenosin, C4^H79N08, consisted of the sugar now shown to be 

 identical with galactose, Q^yfi^, of neurostearic acid, CjgHjgOo. 

 an isomeride of stearic acid, fusing at 84°, and of sphingosine, an 



