The following evidences of the influence of spherical undulations in determining the various harmonies of the solar system, may perhaps interest the readers of the ANALYST: Let Vλ = Vλ = mean velocity communicable by oscillations of V2, the velocity varying between 0 and V2. mean velocity of a perpetual radial oscillation, synchronous with a circular oscillation of V2, and nearly equal the mean velocity of a synchronous very excentric elliptic oscillation. velocity of planetary revolution at Sun's equatorial surface under the volume due to internal work. equatorial velocity of solar rotation under the volume due to internal work mean velocity of a radial oscillation between the two members of our binary star (the Sun and Jupiter) synchronous with the revolution of the binary star about its center of gravity (4332.585 days). 4 V" = mean velocity of a perpetual radial, or extremely excentric oscillation, synchronous with the revolution of the binary star = mean velocity of the binary star in space. T' T"= time of revolution, rotation for V' V". t't"= time of revolution, rotation for Earth. TT time of revolution, rotation for Jupiter. = 2 ratio of the integral of infinitesimal impulses during revolution in a circular orbit π", to the integral of similar impulses during fall from circumference to center of same orbit. 312 π° = Neptune's mean distance from Sun, in units of Earth's mean distance. Saturn's mean distance. 311= Asteroidal mean distance, or twice mean distance of Mars. 3273 = Earth's secular mean perihelion distance. = Mercury's secular mean perihelion distance. 11= major axis of Sun's orbit about center of gravity of binary star. heliocentric distance of linear center of oscillation of secular mean perihelion center of gravity of the binary star. The ratio of V' to V" is found by supposing Sun's radius to vary from r to n2 r. Under such variations In the following table A. represents the theoretical values as estimated from V; B, for Jupiter's distance, 4, C the observed values. For T", C is the mean of the several estimates published by Bianchi and Laugier, Lelambre, Petersen, Sporer, Carrington and Faye. The Sun's annual motion is given in units of Earth's radius vector, C being Struve's estimate. For V', A, B and C are deduced from g on Sun, Earth and Jupiter: The series groups the principal planets into four pairs. The correspondence between the theoretical and observed values is given below in units of Sun's radius. The values of the secular mean apsides are taken from "Stockwell's Memoirs on the Secular Variations of the Orbits of the Eight Principal Planets.” The slight discrepancies in the values of T", V', V", seem to be attributable to Jupiter's mean eccentricity, but they are all within the limits of uncertainty of observation. PERFECT CUBES. BY PROF. W. D. HENKLE, SALEM, OHIO. The following interesting facts in reference to perfect cubes were discovered by the writer about seven years ago: Perfect cubes ending in 1, 2, 3, 4, 5, 6, 7, 8, 9 or 0 followed by 1 have roots ending in 7, 4, 1, 8, 5, 2, 9, 6, 3 or 0 followed by 1. Perfect cubes ending in 1, 2, 3, 4, 5, 6, 7, 8, 9 or 0 followed by 3 have roots ending in 1, 4, 7, 0, 3, 6, 9, 2, 5 or 8 followed by 7. Perfect cubes ending in 1, 2, 3, 4, 5, 6, 7, 8, 9 or 0 followed by 7 have roots ending in 7, 0, 3, 6, 9, 2, 5, 8, 1 or 4 followed by 3. Perfect cubes ending in 1, 2, 3, 4, 5, 6, 7, 8, 9 or 0 followed by 9 have roots ending in 3, 0, 7, 4, 1, 8, 5, 2, 9 or 6 followed by 9. Hence every perfect cube ending in 11 has 71 as the ending of its root; every perfect cube ending in 21 has 41 as the ending of its root; and so on. Thus we see that the last two figures of any perfect cube ending in 1, 3, 7 or 9 may be known if the figure before 1, 3, 7 or 9 is also known. For instance the cube root of the perfect cube 185193 may be known to be 57 because all perfect cubes ending in 93 have roots ending in 57. Thus we may know the cube root of the perfect cube 3,869,893 by knowing only 3,..... 93. The first figure of the root must be 1 and the next two 57. If the series 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 be repeated or made circular it may be seen that the series 7, 4, 1, 8, 5, 2, 9, 6, 3, 0 may be obtained by beginning at 7 in the circular series and counting forward successively to the 7th following figure; in the same way the series 1, 4, 7, 0, 3, 6, 9, 2, 5, 8 can be found by counting forward by threes beginning at 1. To get the series 7, 0, 3, 6, 9, 2, 5, 8, 1, 4 begin with 7 and count by threes. To get the series 3, 0, 7, 4, 1, 8, 5, 2, 9, 6 begin at 3 and count by sevens. It is desirable to obtain some method of getting directly the ten's figure of the cube. The following scheme accomplishes this: In the first case multiply by 7 and reject the left-hand digit when there is one; in the second case multiply 3 and add 8; in the third multiply 3 and add 4; and in the fourth multiply 7 and add 6, and reject as before when necessary. The process may be stated in another way: Putting D as the ten's figure of the cube, and sub-figures as the last figures of the cubes. To find the ten's figure of the root of a cube ending in D, multiply D by 7 and cast out the tens. When the cube ends in Ds, multiply D by 3, add 8, and cast out the tens For instance to find the cube root of 912673: The last figure of the root must be 7; to get the ten's figure multiply 7 by 3, add 8, and cast out the tens or reject the left-hand digit; this gives 9, hence the root is 97. Perfect cubes ending in 1, 3, 5, 7, or 9 followed by 2 { Perfect cubes ending in 2, 4, 6, 8, or 0 followed by 4 have roots ending in {2, 3, 0, 4, or 8 have roots ending in 5, 1, 7, 3, or 91 } followed by 8. } followed by 4. } followed by 6. Perfect cubes ending in 1, 3, 5, 7, or 9 followed by 6 have roots ending in {0; 24, 8, 2, or 6 Perfect cubes ending in 2, 4, 6, 8, or 0 followed by 8 have roots ending in { 6, 2, 8, 4, or 0) followed by 2. In these cases there are alternative series. In the case of D2 the corresponding number of the first of the alternative series can be obtained by multiplying D by 3, and adding 2, and rejecting the tens; in the case of D and D6, by multiplying D by 2, adding 3, and rejecting the tens, and in the case of D ̧, by multiplying D by 3, and rejecting the tens. To obtain the corresponding alternative of the second series subtract 5 when possible, if not add 5. What is the cube root of the perfect cube 110592? Multiplying 9 by 3, adding 2, and rejecting the tens we have 9. The alternative is 5 less, or 4. Hence the root is either 98 or 48. The 110 shows that it must be 48. What is the cube root of the perfect cube 25,412,184? Multiplying 8 by 2, adding 3, and rejecting the tens we get 9. Hence the ten's figure of the root is either 9 or 4. From 25 we get the first figure 2, and from 4 the last figure 4. The root is either 294 or 254. We decide in favor of 294 because 25 is nearly the cube of 3. There is another mode of deciding which is the root: Assume either one as the cube root, subtract it from the cube and the remainder is divisible by 6 if the root assumed is right. A cube when divided by 6 gives the same remainder as when its root is divided by 6. Perfect cubes ending in 2 or 7 followed by 5 |