456 E. D. Preston— Study of the Earth's 



after it. The atmospheric effect has been treated in one 

 system of equations, where the unknown quantities vary 

 directly as certain powers of the pressure and inversely as 

 powers of the temperature. All these corrections are of much 

 more importance in absolute determinations than in relative 

 ones. This brings us to the different methods of observing. 

 Two methods have been chiefly followed. First by noting 

 coincidences between the gravity or experimental pendulum, 

 and the pendulum of a clock set up a short distance away. 

 This is by far the most easy and accurate method of getting 

 the length of one oscillation of the gravity pendulum. The 

 second method is by registering on a chronograph the passage 

 of the pendulum across a fixed point of reference. Forty of 

 these transits suffice to give a mean value, which carries the 

 accuracy of this part of the operation far beyond that attained 

 in deducing some of the other necessary corrections. The 

 probable error of the mean of a chronographic set is only 0'003 

 of a second and when this is divided by 15000, the number of 

 oscillations in one swing, we get an accuracy beyond one 

 millionth of a second. This is all that can be desired, but the 

 method of coincidences is still more accurate while it is much 

 less difficult to observe. We may commit an error of many 

 seconds in the time of a coincidence without vitiating the 

 result. The distinctive feature of the last method is this : 

 when we commit an error of one second in noting the time, 

 we do not change the value of one oscillation in the ratio of 

 this error to the length of the swing, because both pendulums 

 are moving along together. An error in the time of coinci- 

 dence only means that the result will be in error by an amount 

 equal to the ratio, one has gained on the other in the short 

 time between the true coincidence and the one noted, multi- 

 plied by the ratio of the error to the whole period. To illus- 

 trate by a special case, suppose that in 600 oscillations of the 

 clock pendulum, the gravity pendulum loses two oscillations, 

 and suppose that the coincidence was erroneously noted after 

 602 oscillations had been made instead of 600. This error is 

 l/300th of the interval, but far from introducing an error of 

 1/300 in the length of one oscillation , the error is only 1/3 00th 

 of the ratio of the gain of one jDendulum on the other, that is 

 1/300 of 1/300 or say 00001. It is thus seen that the accuracy 

 of the result is a function of the length of time between two 

 coincidences, and that the longer the interval the more accu- 

 rate will the result be given. One might suppose therefore that 

 the coincidence period might be indefinitely long, but there 

 are economic considerations bearing on th^ question. For 

 instance we cannot afford to wait very long for the coincidence 

 because this would entail too much loss of time. Therefore in 



