March 21, 1878] 



NATURE 



401 



and on the linear transformation of the integral / ,- . — 



Prof. Clifford has an excellent paper on the canonical 

 form and dissection of a Riemann's surface. Prof. H, J. 

 S. Smith contributes the conditions of perpendicularity in 

 a parallelepipedal system, and a very interesting presi- 

 dential address on the present state and prospects of some 

 branches of pure mathematics. Mr. Spottiswoode writes 

 on curves having four-point contact with a triply infinite 

 pencil of curves, and Prof. Wolstenholme gives an easy 

 method of finding the invariant equation expressing any 

 poristic relation between two conies. 



LETTERS TO THE EDITOR 



{The Editor does not hold himself responsible for opinions expressed 

 by his correspondents. Neither can he undertake to return, 

 or to correspond with the writers of, rejected 7nanuscripts. 

 No notice is taken of anonymous communications. 



[ The Editor urgently requests correspondents to keep their letters at 

 short as possible. The pressure on his space is so great that it 

 is impossible otherwise to ensure the appearance even of com- 

 munications containing interesting and novel facts. "] 



Trajectories of Shot 



I HOPE you will be able to afford me space lor a few remarks 

 on the following extract from a paper on the Trajectories of 

 Shot, by Mr. W. D. Niven, which appeared in the Proceedings 

 of the Royal Society for 1877. 



Mr. Niven arranges his paper under three heads, calling them 

 the first, second, and third methods. The third method is the 

 one he favourtt, while he endeavours to dispose of the other two 

 in the following terms : — 



"§ II. It will be observed that the two foregoing methods 

 each open with the same equation (a). Now there is a serious 

 difficulty in the use of that equation. Suppose, for example, we 

 were to integrate over an arc of 1°, we should have to use the 

 mean value of k between its values corresponding to the velo- 

 cities at the be{;inning and end of the arc. But we do not know 

 tlie latter of these velocities ; it is the very thing we have to find. 

 The first steps in our work must be to guess at it. The prac- 

 tised calculator can, from his experience, make a very good esti- 

 mate. Having made his estimate, he determines k. He uses 

 this value of k in equation (a), and if he ge ts the velocity he 

 guessed at, he concludes that he guessed rightly, and that he has 

 got the velocity at the end of the arc. If the equation (a) does 

 not agree with him, he makes another guess, and so on till he 

 comes right." 



The case would be indeed hopeless, if all this was quite cor- 

 rect. But I have to inform Mr. Niven that, in all proper 

 cases Vp may be found cucurately from equation (a), and without 

 any "guessing" whatever. Taking Mr. Nivea's own solitary 

 example, I will calculate the value of v^ at the end of an arc, not of 

 1°, but of 3°, and compare my result with his own. The initial 

 velocity, v^, is here 1,400 f.s., and the corresponding value of 

 the coefficient k^, given in my table, is 104 'o. Substitute this 

 value for k in equation (a), given below, and v^ will be found 

 1 291 7 f.s., & first approximation. Now calculate the mean value 

 of /6 between velocities 1,400 and i,29of s. by the help of the table, 

 and it will be found to be equal to 106 3. Substitute this new 

 value of /('in equation (a), and v^ will be found 1289*8 f.s., a 

 second approximation. We must stop here, because if we 

 attempted to carry the approximation further, we should obtain 

 the same value of k, and therefore of v^, as in the second 

 approximation. Mr. Niven finds v^ — 1290 f s. 



Of course in ordinary cases, a calculator, in making his first 

 approximation to v^, would commence by taking a value of k 

 corresponding to a velocity somewhat below the initial velocity, 

 lu this way a better first approximate value of v^ would be 

 found. Thus, again referring to Mr. Niven's own example, I 

 will take a step over an arc of 6°, from a— + 3° to /3 = — 3°. 

 The iniiial velocity is 1,400 f s. I now go so far as to "guess " 

 that the mean value of k vnll correspond to a velocity consider- 

 ably below 1,400 f.s., and take /^ = I07'9, corresponding to a 

 velocity 1,300 f.s. This gives z'/s = i2o8"i, & first approxima- 

 tion. The mean value oik between 1,400 and 1,210 f.s. is now 

 found to be I07"2, which gives vp = 1 209-0 f.s. Mr. Niven 

 obtains 1207 "4 by stepping over two arcs of 3°. If any further 



adjustment was required, proportional parts might be used, 

 seeing that a correction 5-6 = — 07 gives iv^ = + i '8. 



Mr. Niven then proceeds to question the accuracy of what he 

 is pleased to call the "process of guessing," as follows : — 



" It seems to me, however, that this method of going to work, 

 leaving out of account the loss of time, is open to objection in the 

 point of accuracy. For, first there is no method of determining 

 on what principle the mean value of k is found — what manner 

 of mean it is. Again, let us suppose for an instant that the 

 velocity at the end of the arc guessed at, and the value of k, are 

 in agreement ; that is to say, let the equation 



/I,000\o ,_ /I,000\, - d^ k,r, n 1 /x 



1 1^ sec^ ^ — ( -' Y sec^ a = — -[Pa — Pe) — (a) 



\ ^^ / \ Va / XV g^ PI \ I 



hold for the values of v^ and /• used by the calculator. It by no 

 means follows that he has hit on the right value of v^ and k. 



For if he is dealing with a part of the tables in which 



dk 

 d V 



happens to be nearly equal to -3 —•? -i?^^- (i,ooo)3 .^ ,^ ^^ 



vious that there are ever so many pairs of values of v^ and k 

 which will stand the test of satisfying the above equation. Now 

 an examination of Mr. Bashforth's tables for ogival-beaded shot 

 shows that the value of k diminishes as v increases from 1,200 



feet upwards, so that ~ is negative for a considerable range of 

 dv 



values of v which are common in 



It is not at all 



practice. 



unlikely, therefore, that the value for just stated may often 



dz 



be very nearly true ; in which the case the process of guessing 

 becomes extremely dangerous." 



I here observe that Mr. Niven is not entitled to assume that, 

 because two quantities have the same sign, they will therefore be 

 probably often nearly of equal value. Without discussing the 

 value of his test of danger, I have to state that my tabular value 



of — ^, for velocities above 1,200 f.s., lies between o and — o'og. 



Ivp 



I have calculated the numerical values of Mr. Niven's expression 



dk 

 for -— , for shot fired from various guns, from the Martini-Henry 



rifle up to the 80-ton gun, and have always obtained a numerical 

 result so far outside the limits of the tabular value, that, for the 

 present, I conclude that Mr. Niven's condition (whatever may 

 be its value) is 7irjer nearly satisfied in any practical example. 

 But when a practical case is produced where ' ' ever so many 

 pairs of values of v^ and k" differing sensibly, "stand the test 

 of satisfying the above equation" (a), it shall receive my best 

 attention . 



It is well known that the problem of calculating the trajectory 

 of a shot, like so many other practical problems, does not admit 

 of a direct and complete solution. So that all solutions, being 

 approximations, are more or less erroneous. But I feel perfect 

 confidence in the results given by my methods of calculation, 

 because, the smaller the arcs taken at each step, and the nearer 

 the calculated will approach to the actual trajectory. But methods 

 of approximation require to be used with judgment. For instance 

 with the heaviest shot in use, we may take steps of 5° for veloci- 

 ties above 1,100 fs. ; while for small arm bullets arcs of half a 

 degree will be quite large enough. In any case of real difficulty 

 the remedy will be to divide the trajectory into smaller arcs. 



From what I have said it appears that my method of finding 

 the trajectories of shot, when properly applied, is neither a 

 " process of guessing " nor yet " dangerous. " 



Minting Vicarage, March 8 F. Bashforth 



Australian Monotremata 



I AM surprised to find that "P. L. S." (vol.xvi. p. 439), was not 

 aware that the Echidna Tflf-^j^^jjMJ hystrix, is found in N. Queens- 

 laud. For the benefit of your readers I may mention that the 

 Australian Museum possesses a fine specimen of T. hystrix from 

 Cape York. Mr. Armit, of Georgetown, Mr. Robt. Johnstone, 

 and others, have frequently found them in various paits of 

 Queensland. One specimen from Cape York was obtained there 

 by our taxidermist, J. A. Thorpe, in 1867. 



The Platypus {Ornithorhynchus anatinus) is also found in 

 Queensland as far north as the Burdekin at least, perhaps 

 further. 



Tachyglossus, strictly speaking, has no pouch, but the areola 



