434 



NATURE 



[December i, 1921 



declination and equation of time being constant within 

 1/10° or im. 



For computing a very good lo-in. slide-rule was 

 used, which gives results as good as the Almanac 

 hour-angles will allow and good enough for rough 

 comparison with observed durations. (In careful study 

 of the effects of atmospheric changes on sunset dura- 

 tion one would compute each sunset separately, with 

 exact hour-angles, formula (2), and four-place 

 logarithms.) In copying, the slide-rule results are 

 rounded off to whole seconds. 



In a most unfavourable case, winter solstice and 

 latitude N. 60°, 192 1, assuming the data of Table VI., 

 American Nautical Almanac (given on p. 156, 1922), 

 the error of computing may be tested. Computing 

 the hour-angles for the beginning and end of sunset 

 by the sine formula for time sights with declination 

 and equation of time for the moments, and six-place 

 logarithms, they are 42° 15' 16-4" and 43° 59' 14-4" ; 

 the local mean times corresponding are 2h. 47ni, 370s. 

 and 2h, 54m. 330s. ; the duration of sunset is 

 6m. 56os.=4i6-os. 



The mean hour-angle is 43° 7' 150". Using this 

 in the differential formula (2), four-place logarithms 

 gi^'e 415-5+ s., slide-rule 4150+ s. ; so that the dif- 

 ferential formula is in error bv less than is., or less 

 than I per cent. 



By slide-rule computing the differential formula 

 with the hour of Table VI., 2h. 54m., the hour-angle 

 at end is 43° 45', first approximate semi-duration 51-3', 

 final mid-sunset hour-angle 42° 53'+, final computed 

 duration 4i7-5s. This is an error of 1-55. The result 

 for the corresponding date of 1922, given in the table 

 following, 414s., differs on account of a slight differ- 

 ence in the equation of time. 



^ For lower latitudes the error grows smaller, as 

 sin P approaches i and becomes insensitive ; e.g. for 

 the same date in 192 1 and N. 45° six-place logarithms 

 give 22I-5S., four-place 2217+ s., and slide-rule 22i-5s. 

 For latitudes and dates intermediate among those 

 of the table, and for other longitudes, linear inter- 

 polation suffices about as well as it does for the hour 

 of sunset in Table VI., except that the date intervals 

 are much longer ; sunrise durations may also be inter- 

 polated. While the dates are for 1922, the values 

 given should be good, within their limits of error, 

 in other years as well. 



Examination of the table, or of a graph of the 

 same, shows that during the vear for any latitude 

 there are two maxima about the solstices and two 

 minima about the equinoxes. The maxima are 

 sharper than the minima. In low latitudes the 

 summer maximum is less than the winter, due to 

 the smaller semi-diameter of the sun ; but in high 

 latitudes, beginning above N. 52°, this Is reversed, 

 due to the unsvmmetrical and exaggerated effect of 

 refraction, which makes the sines of summer solstice 

 hour-angles less than those of the winter solstice. At 

 the autumnal equinox the duration is somewhat less 

 than at the vernal, due to less semi-diameter ; this 

 holds good for all latitudes where the term " sunset " 

 has any meaning. Elevation above sea-level would 

 have an effect like refraction. 



Since even constant refraction may thus reverse 

 the effects of changing semi-diameter. It Is clear that 

 varying refraction must have considerable effect on 

 the duration of sunset. In studying this effect, which 

 led to the construction of the table, it Is best to obtain 

 the hour-angle directly with a watch ; for when the 

 time-sight formula is used an assumption has to be 

 rnade about refraction, and an error in this assump- 

 tion mav considerablv affect the hour-angle and so 

 the computed duration. Between the tropics this 



NO. 2718, VOL. 108] 



error is of less account than in higher latitudes, but 

 it is eliminated by the use of a watch. 



The Duration of Sunset (in Seconds), 1922. 



Lat. N. o' 10° 20° 30° 35" 40° 45° 50° 52^ 54" 56» 58' 6o» 



jao- I 142 144 152 i63 180 197 220 253 270 292 319 356 406 



" 15 140 142 150 165 177 192 213 243 258 277 300 328 367 



l^eb. I 136 138 146 160 170 184 203 226 239 254 271 292 318 



- '• 15 133 135 142 155 164 176 193 214 225 237 251 267 287 



Mar. 1 131 133 139 152 160 171 186 205 215 225 237 251 267 



.■. 15 129 131 137 149 157 168 182 201 209 219 231 243 258 



April I 129 131 137 149 158 169 182 202 210 321 232 245 261 



.. 15 130 132 138 150 159 171 186 207 216 228 241 255 273 



May I 132 134 141 154 164 177 192 217 229 243 258 378 300 



.. 15 134 136 144 158 169 183 203 230 244 261 281 306 338 



June 1 136 139 146 163 174 192 212 244 262 283 309 345 394 



,, 15 138 140 148 164 178 194 218 252 272 295 327 368 429 



July I 137 140 148 164 176 193 216 251 269 292 322 364 422 



>, 15 136 138 145 161 173 188 209 242 258 278 304 337 383 



Aug. I 133 136 143 156 168 181 201 227 239 256 275 298 329 



» 15 131 133 140 153 162 175 191 214 226 238 254 272 294 



Sept. I 128 131 137 149 158 170 184 204 213 224 237 251 26S 



„ 15 128 130 136 148 156 167 181 199 208 218 229 242 257 



Oct. 1 128 131 137 148 157 168 182 200 208 218 229 242 257 



>, 15 130 132 139 151 159 171 186 205 214 225 237 251 267 



Nov. 1 134 136 143 156 164 178 195 217 228 241 256 274 295 



„ 15 137 139 146 159 171 185 204 230 243 259 277 300 329 



Dec. I 140 143 150 166 177 193 214 245 261 281 305 336 376 



„ 15 142 144 152 168 181 197 220 254 271 293 322 359 411 



Mar. 21 129 131 137 149 157 168 182 200 209 219 230 243 257 



June 21 138 140 148 164 177 194 217 253 273 297 328 370 434 



Sept. 23 128 130 136 148 156 167 181 199 207 217 229 241 254 



Dec. 22 142 145 153 169 181 19S 222 255 273 295 324 362 414 



WiLLARD J. Fisher. 

 49 Langdon Street, Cambridge, Mass., 

 October 23. 



Relativity: Particles Starting with the Velocity of 

 Light. 



I WISH to point out a peculiar property of the motion 

 of a particle in the theory of relativity when the 

 initial speed Is that of light. The result is valid for 

 both the special and the general theory of relativity, 

 but for simplicity I shall consider here only the former 

 case. We start, then, with the Minkowski formula, 

 ds^ = c'^dt- — dx^ — dy- — dz'- This vanishes for the 

 motion of a light-pulse. The interval ds is imaginary 

 when the velocity is greater than that of light, and Is 

 real when the velocity is. less than that of light. 



If the velocity v of the particle is always equal to c, 

 then all the elementary intervals vanish, so that the 

 length-interval between any two positions Is always 

 zero. If, however, the particle merely starts out with 

 the velocity of light and then slows down, so that the 

 acceleration Is negative, the interval between the initial 

 world-point P and any other world-point Q will not 

 be zero, and we raise the following question : What 

 is the ratio between the arc PQ and the chord PQ, 

 or, rather, what Is the limit of this ratio as Q ap- 

 proaches P ? Here, of course, we. mean that both arc 

 and chord are measured by means of the interval 

 formula. If we use ordinary Euclidean measurement, 

 of course the limit of this ratio for real curves is 

 always unity. For the Minkowski geometry it turns 

 out that the limit is unity whenever the initial velocity 

 Is less than the velocity of light, but In the excep- 

 tional case we are now considering we find that the 

 limit is actually different from unity, and, in spite 

 of ordinary Intuition, is actually less than unity. 



Our precise theorem Is as follows : — // the initial 

 velocity is that of light and the initial acceleration is 

 not zero, then the litnit of the arc to the chord is f V'2, 

 which is api)roxiinately 094. 



If the Initial acceleration Is zero, so that the velocity 

 remains that of light for neighbouring points, then 

 the limit mav have any value. 



That the limit of the arc to the chord Is not always 

 unity for curves in the Euclidean plane was first 



