In gravitational units of mass, based on the metre and second

as units of length and time,
⁠Log. earth's mass = 14.60052
⁠Log. moon's mass = 12.6895.

The sum of the corresponding numbers multiplied by 332600 gives
⁠Log. sun's mass = 20.12773.

Putting a for the mean distance of the earth from the sun, and n for its mean motion in one second, we use the fundamental equation
⁠a²n² = M₀M′,
M₀ being the sun's mass, and M′ the combined masses of the earth and moon, which are, however, too small to affect the result. For the mean motion of the earth in one second in circular measure, we have

⁠n = π/31558149; log. n = 7.29907

the denominator of the fraction being the number of seconds in the sidereal year. Then, from the formula

⁠a² = M₀/n² = [20.12773]/—15.59814

we find
⁠Log. a in metres =   11.17653
⁠Log. equat. rad. ⨁    6.80470

⁠Sine ☉'s eq. hor. par.     5.62817
⁠Sun's eq. hor. par.⁠   8.762″.

IV. The determination of the solar parallax through the parallactic inequality of the moon's motion also involves two elements—one of observation, the other of purely Motion of
Moon. mathematical theory. The inequality in question has its greatest negative value near the time of the moon's first quarter, and the greatest positive value near the third quarter. Meridian observations of the moon have been heretofore made by observing the transit of its illuminated limb. At first quarter its first limb is illuminated; at third quarter, its second limb. In each case the results of the observations may be systematically in error, not only from the uncertain diameter of the moon, but in a still greater degree from the varying effect of irradiation and the personal eqtiation of the observers. The theoretical element is the ratio of the parallactic inequality to the solar parallax. The determination of this ratio is one of the most difficult problems in the lunar theory. Accepting the definitive result of the researches of E. W. Brown the value of the solar parallax derived by this method is about 8.773″.

V. The fifth method is, as we have said, the most uncertain Motion of
Earth. of all; it will therefore suffice to quote the result, which is

${\displaystyle \pi }$ = 8.8l8".

The following may be taken as the most probable values of the solar parallax, as derived independently by the five methods we have described:—
⁠From measures of parallax . 8.802″
⁠From velocity of light . .         8.781″
⁠From mass of the earth . .     8.762″
⁠From par. ineq. of moon . .    8.773″
⁠From lunar equation . .⁠   8.818″

The question of the possible or probable error of these results is one on which there is a marked divergence of opinion among investigators. Probably no general agreement could now be reached on a statement more definite than this; the last result may be left out of consideration, and the value of the solar parallax is probably contained between the limits 8.77″ and 8.80.″ The most likely distance of the sun may be stated in round numbers as 93,000,000 miles.  (S. N.) 

PARALLELISM, PSYCHOPHYSICAL, in psychology, the theory that the conscious and nervous processes vary concomitantly whether or not there be any causal connexion between them; in other words “ that modifications of consciousness emerge contemporaneously with corresponding modifications of nervous process ” (Stout). The theory is the third possible alternative in considering the relation between mind and body, the others being interaction and one-sided action (e.g. materialism). It should be observed that this theory is merely a statement, not an explanation. (See Psychology.)

PARALLEL MOTION, a form of link-work invented by James Watt, and used in steam-engines (see Steam-Engine, § 88), to connect the head of the piston rod, moving up and down in a vertical path, with the end of the beam, moving in the arc of a circle. An ordinary form is shown diagrammatically in

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Watt's Parallel Motion.

figure. MN is the path in which the piston-rod head, or cross head, as it is often called, is to be guided. ABC is the middle line of half the beam, C being the fixed centre about which the beam oscillates. A link BD connects a point in the beam with a radius link ED, which oscillates about a fixed centre at E. A point P in BD, taken so that BP: DP:: EN: CM, move in a path which coincides very closely with the straight line MPN. Any other point F in the line CP or CP produced is made to copy this motion by means of the links AF and FG, parallel to BD and AC. In the ordinary application of the parallel motion a point such as F is the point of attachment of the piston-rod, and P is used to drive a pump-rod. Other points in the line CP produced are occasionally made use of by adding other links parallel to AC and BD.

Watt's linkage gives no more than an approximation to straight-line motion, but in a well-designed example the amount of deviation need not exceed one four-thousandth of the length of stroke. It was for long believed that the production of an exact straight-line motion by pure linkage was impossible, until the problem was solved by the invention of the Peaucellier cell. (See also Mechanics: Applied Mechanics, §§ 77, 78.)

PARALLELS, in siegecraft, a term used to express the trenches drawn by besiegers in a generally parallel direction to the front of a fortress chosen for attack. Parallels are employed along with “ zigzag approaches ” in the “ formal attack ” or siege proper. They are traced in short zigzag lengths (the prolongation of each length falling clear of the hostile works), in order to avoid enfilade; but their obliquity is of course made as slight as is consistent with due protection in order to save time and labour. The “ first parallel ” is opened at a convenient distance from the fortress, by numerous working parties, who dig (under cover of night) a continuous line of entrenchments facing the point or points of attack. Zigzags are next dug to the rear (when necessary) to give sheltered access to the parallel, and from this new zigzags are pushed out towards the defenders, to be connected by a “ second parallel,” and so on until finally a parallel is made sufficiently close to the fortress to permit of an assault over the open, the parallels becoming stronger and more solid as they approach to closer range. This system of parallels provides, within range of the defenders' -ft'eapons, shelter in which the besieger can safely mass men and material for the prosecution of the attack. Parallels and approaches are constructed either by ordinary “ trench work,” executed simultaneously by a large number of men strung out along the intended line, or by “ sapping ” in which one trained “ sapper,” as it were, burrows a trench in the required direction, others following him to widen and improve the work.

PARALUS and SALAMINIA, the name of two ancient Athenian triremes used for sacred embassies, the conveyance of dispatches and tribute money, the transportation of state criminals, and as flagships in time of war. It is probable that a third vessel of the same kind (called Delia) was used exclusively for Delian embassies, although it has been identified by some with the Salaminia.

PARALYSIS, or Palsy (from Gr. παραλύειν, to relax; Wycliffe has palesy, and another old form of the word is parlesy), a term which in its wider acceptation indicates abolition of motor, sensory, sensorial or vaso-motor functions, but in medical nomenclature is usually restricted to the loss or impairment of voluntary muscular power. Paralysis is to be regarded rather as a symptom than a disease per se; it may arise (1) from injury