5o6 



NATURE 



[April 2^, 1878 



tion as to how far it is applicable to the data submitted for 

 discussion. 



Dr. Hunter published the data for discussing the rainfall at 

 Madras during the six sun-spot cycles, ending 1876, these being 

 all the available data for Madras. As regards the sun-spots, we 

 certainly have no positive data earlier, at least, than these cycles, 

 whatever value may be attached to the approximate earlier 

 figures supplied by Dr. Wolf. As regards, therefore, both the 

 elements under discussion, viz., the sun-spots and the rainfall, the 

 period discussed by Dr. Hunter represents the whole of the 

 cycles for which material is available. 



In dealing with this period. Dr. Hunter divides it into six 

 equal cycles of eleven years each, this being substantially the 

 average duration of the sun-spot cycles. I have an-anged the 

 relative numbers published in Wolf's last list ( Wolf. Astrono- 

 mische Miitheilungen, pp. 35-37), according to the cycles adopted 

 by Dr. Hunter, with the result that all the six minimum years of 

 sun-spots occurred either in the first year of the cycle, or in one of 

 the immediately adjoining ones on either side of it, viz., in the 

 second or in the eleventh years. As regards the years of maximum 

 sun-spot, five out of the six occurred in the fifth or sixth years of 

 the cycle, and the remaining year of maximum sun-spots occurred 

 in the eighth year. 



In his paper Dr. Meldrum states that as the sun-spot cycles 

 are not all of the same length, it is evident that, by starting 

 from any one year and going backwards over a long period, 

 always using the same fixed number, a maximum and a minimum 

 year might fall into the same group, and it was to obviate the 

 occurrence of this contingency which the above analysis of Dr. 

 Hunter's method shows did not occur during the period discussed 

 by him, that Dr. Meldrum has proposed his new method as a 

 more accurate mode of discussing the data. 



To test the value of this new method of inquiry, I have 

 arranged Wolf's relative numbers of sun-spots in accordance 

 therewith, the maximum year of sun-spots of each cycle being 

 placed in the sixth year, the minimum years being marked with 

 an asterisk, and the "mean cycle" of eleven years being calcu- 

 lated from the thirteen years in the manner described by Dr. 

 Meldrum :— 



It will be seen from this table that with this arrangement the 

 year of minimum sun-spots has occurred on the tenth, twelfth, 

 thirteenth, first, second, and third years. By Dr. Hunter's 

 arrangement the minimum years fell within a compact group of 

 three consecutive years out of a cycle of eleven, whereas by Dr. 

 Meldrum's arrangement they are scattered over seven years out of 

 a cycle of thirteen. Further, I find that in the second cycle 

 what is virtually a maximum year (viz., 1836 with Ii9'3 of 

 sun-spots) fell within his minimum group, or in the thirteenth 

 year. This is precisely the result which the method was designed 

 to avoid, but as to the occurrence of which there was not an 

 approach under Dr. Hunter's arrangement. 



Again, if the same relative numbers of Wolf be arranged as 

 Dr. Meldrum proposes, so that the year of minimum sun-spots 

 of each cycle be placed in the ninth year of the thirteen years, 

 it will be found that the maximum years are scattered over the 

 twelfth, thirteenth, first, second, third, and fifth years of the 

 series. By Dr. Hunter's method of arrangement five out of the 

 six maximum years fell in the fifth and sixth years of the series, 

 while the remaining one fell in the eighth year, thus again pre- 

 senting a compact group, whereas Dr. Meldmm's method 

 scatters them over more than half of his series of thirteen 

 years. 

 An objectionable feature of this new method is the necessary 



repetition of figures which it involves. Thus, in the table given 

 above, embracing six cycles, nine minimum years occur ; and in 

 the table in which all the minimum years are so arranged as to 

 stand in the ninth year of the cycle, nine maximum years also 

 occur, so that if the Madras rainfall were discussed by this 

 method, the averages would be computed from tables in which 

 the maximum and minimum years occur eighteen instead of 

 twelve times. 



Mr. Meldrum's method might be improved if he entirely 

 struck out the first and thirteenth years of the thirteen years 

 series, and simply "bloxamed" the remaining eleven years for 

 the years of his " Mean Cycle;" that is, made the first of 

 these years the mean of the eleventh, first and second ; the 

 second year the mean of the first, second and third. Even, 

 however, with this change the method is inferior to that 

 employed by Dr. Hunter, and the force of this statement will 

 be the more readily recognised if it be kept in mind that we 

 have no positive data from which the relative numbers of the 

 sun-spots can be calculated prior to the time when Schwabe 

 began his great work of sun-spot observation. 



Edinburgh, April 22 Alexander Buchan 



Trajectories of Shot 



Mr. Niven was perfectly welcome to make use of my experi 

 ments and tables, as he has done, in trying to devise new methods 

 of calculating trajectories of shot. And when he had satisfied 

 himself that his methods possessed some advantages over others, 

 he required no excuse whatever for their publication. But I 

 altogether object to Mr. Niven's rule for finding v^ being con- 

 nected in any way with the mode of calculation adopted by me. 

 I beg, therefore, to place side by side Mr. Niven's rule, to which 

 I object, and my rule, which I make use of, and so leave the 

 matter. Mr. Niven says respecting v^ : — 



* ' The first steps in our work must be to guess at it. The 

 practised calculator can, from his experience, make a very good 

 estimate. Having made his estimate he determines k. He 

 uses the value of k in equation (a), and if he gets 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 equation (a) does 

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

 comes right." 



The following is the course I pursue to find vp. Refer to the 

 table of coefficients and take out the value of ka corresponding 

 to the initial velocity Va.. Substitute in equation (a) and find a 

 first approximate value of v^. Now determine the mean value 

 of k between v^ and vp just found, substitute in equation (a), and 

 thus find a second approximate value of v^, which will generally 

 be found sufficient. Otherwise adjust by proportional parts. 



In this way the value of v^ is found accurately on the supposi- 

 tion that k has remained constantly at its mean value between 

 v^ and vp. Here the operations are of the simplest kind, and 

 no guessing or practised calculator is required. And with a 

 view to diminish the tedium of making these calculations, tables 

 of 2 {k), 2 (/^ -T- g), (1000 -f- v)^„ &c., have been calculated and 

 printed, but their publication has been delayed on account of 

 the experiments proposed to be made with low velocities. 



Since Mr. Niven described the process of guessing as ^^ ex 

 tremely dangerous," there can be no doubt that the epithet was 

 "extreme." As I supposed, he is not prepared to supply me 

 with a single practical case where his condition of danger is 

 satisfied. And if a case cannot be found then the objection falls 

 to the ground. Whether we consider the range of values of k 

 for spherical or ogival-headed shot, for velocities above 1,200 f .s., 



we shall find that — - lies between the limits and r- o'OQ, or, 

 dv 



where ^ is a mean over an]arc, between and - 0*05 about. And 

 it is the smallness of this tabular value which renders it difficult, 

 if not impossible, to satisfy Mr. Niven's condition of danger. But 

 if this quantity had not been small, then the cubic law could not 

 have been used even approximately. Mr. Niven is at liberty 

 to take shot of any size used in practice, moving at any attain- 

 able velocity beyond 1,200 f.s., and the coefficients of resistance 

 for either spherical or ogival-headed projectiles. The objection 

 is Mr. Niven's, and he must take the onus of supporting it if 

 he still thinks it of value. 



I regret to have to write anything in opposition to Mr. Niven's 

 paper, because in all other respects it appears to me a valuable 

 contribution to the science of ballistics, F. Bashforth 



Minting Vicarage, April 1 7 



