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Amplissimis Curatoribus, Rectori Magnifico, Doctissimoque Senatui Universitatis Upsaliensis.

Societas Regia Edinensis nos jussit, viri illustrissimi, vobis hoc sollemni die gratulari, quo nihil exoptatius nobis evenire potest. Itaque nobis animo perpendentibus quam excultus studiorum status hodie sit, et quantopere vos in his augendis et amplificandis excellueritis, grato animo laudanda est illa fructuosissima opera quam vos et Academia vestra in finibus scientiarum dilatandis posuistis; et admirationem observantiamque nostram in viros præclaros qui temporibus præteritis exornaverunt et in successores eorum qui nunc exornant Academiam vestram libenter enuntiamus. Ab Upsala Societas nostra Regia semper lumen exspectavit et accepit; in Astronomia, Meteorologia, rebus Botanicis et Physicis, Historia Naturali, reliquisque studiis Academicis, in quæ Socii nostri continuo incumbunt, problemata difficillima solvistis, naturæ recondita extricavistis, patefecistis et illustravistis; et vestro proventu secundo Socii nostri in difficultatibus superandis valde profecerunt. In regionibus Septentrionalibus juventutem disciplinis humanis et subtilioribus instituendo Universitas Upsaliensis de republica literaria optime meruit, et laboribus suis philosophicis de rerum natura orbi terrarum notissimis fons et origo luminis fuit, imo quidem aurora borealis, vel potius sol meridianus, Sueciæ et aliarum nationum. Ut gloria vestra per omnia sæcula permaneat ex imo pectore optamus et precationibus ominamur.

JOANNES HUTTON BALFOUR, M.D., Societatis Regiae Edinensis Secretarius.

Nonis Sextilibus,

ANNO CHRISTI MDCCCLXXVII.

3. On a Method of Determining the Cohesion of Liquids. By J. B. Hannay, F.C.S.

(Abstract.

In this paper the author concludes that the measurement of the breaking-strain of liquids is the only universally applicable method of measuring their cohesion; and as dropping is the phenomenon in which the breaking-strain can be most easily measured, he examines the work already done in this direction. Dr Guthrie's theory that,

the increase in the size of drop, with the increase in the rate of dropping, is due to the attraction of the solid tearing more of the root of the drop in low than in high rates is put to a crucial test by an experiment in which mercury drops from a wide glass tube so arranged that the tube only acts as a support for a column of liquid from the end of which the drops fall. In this case there is no solid to reclaim by adhesion any of the drop, and yet there is the same increase in size as the rate increases. The author accounts for the increase as follows:-1. The rupture of the neck of a drop is not an instantaneous process, but lasts for a short time, and during that time liquid is flowing into the drop through the neck, and the faster the flow the greater is the increment of the drop during rupture. 2. When the rate is high the breaking neck has a longer life-time, as the stump follows after the full drop as in the beginning of the formation of a stream. Briefly stated, the quicker the rate the larger the drop, because more liquid flows into the drop after the rupture has commenced, and the longer does that flow continue. The author calls a drop when it begins to break a "normal" drop; and to find its weight he determines the decrease of weight with decrease of rate, and reduces the latter to zero when the weight of a normal drop is found. Apparatus is shown by which experiments were carried out, and inaccuracies eliminated. The normal drop is found to weigh 0-4130 grm., and as the width of the neck is found to be 3.395 mm., this gives a breaking strain of 0.0456 grm. per square millimeter for mercury at 16° C.

4. Note on Vector Conditions of Integrability. By Professor Tait.

(1.) The relation

P

do = uqdpq1

ensures that the tensor of do shall always be u times that of dp Hence, if be the common vector of three series of surfaces which together cut space into cubes, possesses the same property. (See § 6 of my paper On Orthogonal Isothermal Surfaces, Trans. R.S.E., 1873-4. In what follows this paper will be referred to as .) We may suppose the tensor of q to be any constant, unity suppose. Then, from

VOL. IX.

Tq2 = 1,

4 A

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From the three equations of this form we obtain by the operations S.i, S.j, S.k, nine scalar equations, of which the following are three :

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The last of these, with its two similar equations, shows that

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which express Dupin's theorem for this particular case.

(3.) If we put for simplicity

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the equations of last section give at once three like

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(4.) But we have, by differentiation, from the second equations

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(5.) But if, instead of combining the last set of three we equate to zero the scalar coefficients of i, j, k separately in each, we have three equations of each of the following forms :

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where the right hand members are functions of x, y, z respectively.

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and the first of the second set of three equations becomes

or

u3(X" + Y′′ + 2uX'2+2uY′2) = u1(X'2 + Y′2 — Z′2),

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or, as we may take the origin where we please,

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This is, therefore, the only value of which satisfies the conditions of the problem, and the last equation in § 4 above shows that either C or D must vanish. If C vanish, u and q are both constant.

(6.) If D vanish, we have by § 3 above

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so that is the Electric Image of p rotated through any angle about any axis through the centre of the reflecting sphere. (§12.) (7.) If the equations of any three systems of orthogonal surfaces be

F1 =C1, F2=C2, F3=Cg,

we may obviously write for the flux of heat through each the expressions

VF1 = u1giq1, VF2=u2qiq ̃1‚ ▼F2 = uçqkq ̃' ;

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