171 



OSCILLATION. 



OSIKIS. 



where r stands for tin ' . no f. To find the time of a Mini-vibration, 



to f=0, 



this must be integrated from V = ' to f = 0, or from $ - 



that is, after change of sign, from + =0 to f = -j w. Now between these 



limits 



.. . 1.8.8 2 - ' r 



/n *-<f* = 2 . 4 . 6 .... 2l , -2 



whence, changing the sign of the preceding and integrating each term 

 by this formula, we find the time of a semi-vibration. Double this, 

 to find the time of a vibration, which call T, and we have 



'lr 1 1.9 



- . -I 1 + -. sin 1 o' + -T- r.- sin 4 o' + 



1.9.25 I 



O6T86 ^- a + ' ' ' } 



a rery convergent series. If a' be small, we have, with an error of the 

 second order only. 



or the time of vibration of the same pendulum in different small arcs 

 it very nearly independent of the lengths of the arcs ; a result which 

 might be obtained from the considerations given in the article 

 ISOCHRONISM. To take in two terms cf the series is very nearly 



equivalent to multiplying the preceding value of T by 1 + y^o 1 . 



The number of vibrations in a mean solar day of the pendulum 

 whose length is I, placed in a vacuum, is 



--* 



Otcillation, centre of. Let a number of material points invariably 

 connected together vibrate about a horizontal axis. It is required to 

 find at what distance a single material point must be hung that it may 

 vibrate in the same manner. 



Let there be a number of material points, or infinitely small bodies, 

 having the masses m, m', m", &c., invariably connected with each 

 other, and with an horizontal axis of rotation, their perpendicular dis- 

 tances from the axis being /, I', I", Ac. Let M = m + m' + , &.C., be the 

 sum of all the masses, and let k be the perpendicular distance of the 

 centre of gravity of the whole from the axis. When the last-men- 

 tioned perpendicular distance makes an angle * with the vertical, the 

 moving force is M </ acting in the direction of gravity at the distance 

 X- xin 8, and acting with a moment of rotation M;/t sin i. Let $ be 

 the angular velocity at the time in question, which becomes 4> + c/<fi at 

 the end of a new interval d t ; then taking the mass m into consideration, 

 we have Ity for its actual velocity, and I d <p for the increment in 

 the time d t, whence m I d <p is the actual momentum gained, 



m I -j- the moving force which would produce that momentum. This 



force acting at the distance I, and perpendicularly to that distance, 



d<f> 

 would exert a moment of rotation m P -^ . Ascertain in the same 



way the moments of rotation of the other masses ; then by the equi- 

 valence of the impressed and effective forces [VIRTUAL VELOCITIES] 

 we must have 



8. 



d$ 



a v a o 



But t> is g-j, thus written since g- is negative ; if then we denote 



*<* m'l f> + ...(which is the MOMENT OF INERTIA) by 2roP, we 

 readily deduce from the preceding 



d*8 Mt 



dt' + lml'- 



The question of the motion of this system is now completely reduced 

 to that of a simple pendulum ; for if we compare the preceding 

 equation with that of the motion of a simple pendulum, we find that 

 the two would agree if the length (there called /) of the simple pen- 



2m/ 1 

 dulum were ~ M ~j" or : any rigid system which makes oscillations 



about an horizontal axis, oscillates in the aame manner as a simple 

 pendulum, the length of which is the moment of inertia of the system 

 with respect to it* axis divided by the product of the whole mass and 

 the distance of the centre of gravity from the axis. If in the line X- 



2m/* 

 or its continuation, a distance equal to ~~jf be sot off from the axis, 



the moving extremity is called the centre of oscillation, as being that 

 point in which the whole mass might be collected without any alteration 

 of the law of oscillation. 

 Let o be the moment of inertia when the axis passes through the 



centre of gravity and is parallel to the given axis ; then [MOMENT 

 lN>:RTiA]2m/*=a + M*', whence the distance of the centre of onci 



or 

 centre of oacilLv 



tion from the axis is 



,or JT+fc Hence the centre of oscilla- 



tion is always farther from the axis than the centre of gravity by 

 o 

 . Let this be called h : we have then 



Now a and M are independent of the position of the axis, o depend- 

 ing only on the masses and manner in which the masses are dis- 

 tributed about the centre of gravity, and M on the amount of the 

 iniKmnn, If then a new axis of suspension were taken, distant by A 

 from the centre of gravity on the other side, that is, if a new axis of 

 suspension were taken passing through the first centre of oscillation, 

 and if A' were the distance of the new centre of oscillation from the 

 centre of gravity, we should have 



*'* = M ' but * * = "M ' wnence *'=* = 



that is, the old axis of suspension contains the new centre of oscilla- 

 tion, if the new axis of suspension contains the old centre of oscillation. 

 This is generally expressed thus : the centres of suspension and 

 oscillation are convertible. When the body is a continuous mass, the 

 preceding investigations apply, but the integral calculus must be 

 employed to determine u. 



OSCULATION, a term used instead of contact by foreign writers. 

 [CONTACT ; TANGENT.] 



OSCULUM PACIS. [PAX.] 



OSI'RIS, one of the principal Egyptian deities, was the iirother of 

 Isis and the father of Orus [Isis ; Onus], and is said by many writers 

 to have been the first king of Egypt His history is given in the first 

 book of Diodorus, and in the treatise of Plutarch, ' On Isis and Osiris ;' 

 but it is probable that the genuine Egyptian traditions respecting this 

 deity had been considerably corrupted at the time of these writers. 

 According to their accounts however, Osiris was the first who reclaimed 

 the Egyptians from a state of barbarism, and taught them agriculture 

 and the various arts and sciences. After he had introduced civilisation 

 among his own subjects, he resolved to visit the other nations of the 

 world, and confer on them the same blessing. He accordingly com- 

 mitted the administration of his kingdom to Isis, and gave her Hermes 

 to assist her in council and Hercules to command her troops. Having 

 collected a large army himself, he visited in succession Ethiopia, 

 Arabia, and India, and thence marched through central Asia into 

 Europe, instructing the nations in agriculture and the arts and sciences. 

 He left his son Macedon in Thrace and Macedonia, and committed the 

 cultivation of the land of Attica to Triptolemus. After visiting all 

 parts of the inhabited world, he returned to Egypt, where he was 

 murdered soon after his arrival by his brother Typhon, who cut up his 

 body into twenty-six parts, and divided it among the conspirators who 

 assisted him in the murder of his brother. These parts were after- 

 wards, with one exception, discovered by Isis, who enclosed each of 

 them in a statue of wax, made to resemble Osiris, and distributed them 

 through different parts of Egypt. ThU myth appears to allude to the 

 fact mentioned by Herodotus, that Osiris was the origin of the mummy 

 form. After leaving this world Typhon, or the evil principle, waa 

 overcome by his influence. For the various explanations of the myth 

 of Osiris, see Wilkinson, ' Ancient Egyptians,' iv. 333, &c. 



Both ancient and modern writers have differed considerably respect- 

 ing the peculiar attributes and powers of this deity. He appears to 

 have been the representative of the supreme beneficence. The worship 

 of Osiris consequently was universal in Egypt though, says Herodotus, 

 it was not customary for all the Egyptians to worship the same god, 

 every one paid adoration to Osiris and Isis. Wilkinson says that 

 Osiris differed from most of the other Egyptian deities, inasmuch as 

 they were regarded as human beings who were deified after death, 

 whereas Osiris was a manifestation of deity in a human form ; and 

 this was " the profound secret revealed only to some of those who 

 were initiated into the highest order of the mysteries." (' Ancient 

 Egyptians,' iv. 317.) Many of the ancients believed that he repre- 

 sented the sun or the Nile ; while his discovery of the vine and his 

 expedition to India led others to identify him with Dionysus. (Herod., 

 ii. 144.) Herodotus informs us (ii. 48) that the festival of Osiris was 

 celebrated in almost the same manner as that of Dionysus. It appears 

 however not improbable that the worship of Osiris was introduced into 

 Egypt, in common with the arts and sciences, from the Ethiopian 

 Meroe. We learn from Herodotus (ii. 29) that Zeus (Atuinon) and 

 Dionysus (Osiris) were the national deities of Meroe ; and we are told 

 by Diodorus (iii. 3) that Osiris led a colony from Ethiopia into Egypt. 



Osiris was venerated under the form of the sacred bulls Apis and 

 Mnevis (Diod. i. 21); and as it is usual in the Egyptian symbolical 

 language to represent their deities with human forms and with the 

 heads of the animals which were their representatives, we find statues 

 of Osiris represented with the head or the horns of a bull, accompanied 

 with the name Apis-Osiris (Wilkinson, iv. 347). 



Osiris, in common with Isis, presided over the world below ; hU 

 principal office as an Egyptian deity, was to judge the dead, and he is 



