THE PATH OF A ROTATING SPHERICAL PROJECTILE. 



497 



and I took as initial data F=300, <p=15° ; [whence, of course, F 2 = 84,000 nearly. 

 This is required for comparison with the exact solution.] 



Working from these he obtained a mass of results from which I make a few 

 extracts : — 



8/6 



1/v S(l/») 



4> 



sin <f> 2(sin <£) cos <£ 2(cos </>) 



1. 



2. 

 3. 



90,000 



85,401 



81,032 

 * 



300 



292'2 



284-6 



•003 



•00342 



•00351 

 * 



•003 



•00675 



•01026 



15° 



14-876 



14-746 

 * 



•2588 

 •2565 

 •2546 



•2588 

 •5153 



•7699 



* 



•9659 

 •9665 

 •9671 



•9659 

 1-9324 

 2-8995 



20. 

 21. 



33,045 



31,319 

 * 



181-8 

 177-0 



•00550 

 •00565 



•08666 

 •09231 



11-028 



10-686 

 * 



•1914 



1854 



4-6102 



4-7956 



* 



•9815 

 •9826 



19-4569 

 20-4395 



■X- 



40. 

 41. 



11,440 



10,875 



* 



106-9 

 104-3 



•00935 



•00959 



■X- 



•23391 

 •24350 



- 1-023 - 



- 2 030 - 



* 



0178 

 0355 



6-6163 



6-5808 

 * 



•9998 

 •9994 



39-3178 



40-3172 



■it 



60. 

 61. 



5453 



5377 

 * 



73-8 

 73-3 



•01354 



•01363 



* 



•46935 



•48298 



- 30-748 - 

 -32-564 - 



* 



5113 

 5383 



1-4677 

 •9294 

 * 



•8595 

 •8428 



58-3988 



59-2416 

 * 



This table gives simultaneous values of s, v, and (p directly, t is obviously to be 

 found by multiplying by 6 feet the numbers in column fifth ; while by the same process 

 we obtain rectangular coordinates, vertical and horizontal, from the eighth, and the last, 

 columns respectively. Thus for instance we have simultaneously 



s 



V 



t 



4> 



y 



X 



120 



181-8 



9 -52 



ir-028 



27-66 



116-74 



240 



106-9 



1-404 



- 1 023 



39-69 



235-9 



(The trajectory is given as fig. 3 in the Plate, and will be further analysed in the 

 next section of the paper.) 



From the complete table we find that, in this case, <p is positive up to the 38th 

 line inclusive, and then changes sign. It vanishes for s = 233 (approximately) after the 

 lapse of l s, 35. The rectangular coordinates of the vertex are about 230 and 40, and 

 the speed there is reduced to 110. From the exact equation we find s = 232 for (f> = 0°. 

 This single agreement is conclusive, since the earlier tabular values of s for a given 

 value of (p ought to be somewhat in excess of the true values ; while the later, and 

 especially those for negative values of (f> greater than 30° or so, should be somewhat too 

 small : — i.e. the calculated trajectory has at first somewhat too little curvature, but 

 towards the end of the range it has too much. It is easy to see that this is a necessary 

 consequence of the mode of approximation employed : — look, for instance, at the fact 

 that the initial speed is taken as constant through the first six feet. See also the 

 remarks in § 17. On the whole, therefore, though the carry may possibly be a little 

 underrated, the numerical method seems to give a very fair approximation to the truth. 

 This admits of easy verification by the help of the value of d(f>/ds last written, for it 

 enables us to calculate the exact value of s for any assigned value of <p by a simple 

 difference calculated from the result obtained from an assumed value. 



19. Taking the method for what it is worth, the following are a few of the results 



