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THE SURVEYOR.

The Journal of the Institution of Surveyors, N. S. Wales

(INCORPORATED).

VOL. XIII. No. S.

SYDNEY, AUG 28, 1900.

PRICE, 6D.

"THE SURVEYOR," which is published monthly, contains original articles on matters connected with Surveying, both of a technical and of a general character, extracts from Scientific journals on allied subjects, notes upon topics of professional interest, notices of appointments or changes in the personnel of the Government Surveying and Engineering Staffs, reports of the proceedings and transactions of the Scientific Institutions, and correspondence.

Literary matter may be sent to the Editors, T. H. Loxton and John Miller, at the Office of the Institution, Queensland Offices, Bridge Street, Sydney.

Business communications should be addressed to the Business Manager, E C. Hughes, Queensland Offices, Bridge Street, Sydney.

Items of News and Criticisms upon Contributed Articles appearing in our columns will be especially welcome.

The Editors do not hold themselves responsible for the opinions of correspondents, nor for the return of manuscript.

GENERAL.

APPOINTMENTS, PROMOTIONS, ETC., IN THE PUBLIC SERVICE.—Mr. Victor Gordon Fisher has been appointed a Cadet Draftsman, Lithographic Branch, Department of Lands; and Mr. John James Baker, a Cadet Draftsman in the District Survey Office, Maitland. Mr. Licensed Surveyor H. K. W. Mackenzie has been appointed a Mining Surveyor.

GENERAL MEETING. The ordinary monthly General Meeting of the Institution of Surveyors, N.S.W., was held at the Rooms,_57 Queensland Offices, Bridge-street, on Tuesday, 14th inst., at 8 p.m. The chair was taken by the President, Mr. G. H. Knibbs, and there were present Messrs. E. A. Bonney, S. Mills, H. F. Halloran, C. C. Bullock, W. Jahn, T. H. Loxton, Lee, Dobbie, Henwood, Arnheim, Miller, Cooke, and several visitors.

The minutes of previous meeting having been read and confirmed, a ballot was taken, and Mr. Henry St. John Wood, District Surveyor, elected a member, and Mr. George Heimbrod an associate.

Mr. T. F. Furber then read an interesting paper on Map Projection," which will be publishsd in our next issue. After some remarks by the President on the subject matter of Mr. Furber's paper, the meeting adjourned, the discussion on the present trend of the professional activity of surveyors being postponed till next meeting.

PERSONAL.-We note that Mr. C. A. Schott, who has been for so many years identified with the work of the United States Coast and Geodetic Survey, the computing division of which he had charge of, has retired in order to devote his whole time to certain scientific work. has been succeeded by Mr. J. F. Hayford.

He

NEW DYNAMOMETER.--A new form of instrument is now being advertised in Nature, Rintoul's Spring Dynamometer, which may very easily be an improvement on the Salter's Balance in general use among surveyors for measuring the tension of their steel bands. We shall be glad to know if any of our readers have yet tried it.

SURVEYORS FOR NEW GUINEA.-We understand that it is probable several surveyors will shortly be required for employment under the Government of British New Guinea, and that the salary will probably be about £400 per annum.

APPOINTMENTS, PROMOTIONS, ETC., GOVERNMENT SERVICE.-Mr. G. W. Conolly has been transferred from the position of Acting District Surveyor at Bourke and appointed District Surveyor at Armidale. Mr. Staff-Surveyor E. J. Coberoft, of Maitland, has been appointed Acting District Surveyor at Bourke.

A1

VOLUMES OF SOLIDS.

PAPER entitled "On the relation, in determining the volumes of solids, whose parallel transverse sections are nic functions of their position on the axis, between the number, position, and co-efficients of the sections, and the (positive) indices of the function," was read at the Royal Society by Mr. G. H. Knibbs, F.R.A.S., Lecturer in Surveying, University of Sydney, on 6th June last, 1900. A copy of this, pp. 36-71, Journ. Roy. Soc., Vol. xxxiv., has been presented by the author to the Institution of Surveyors

The object of the paper may be thus defined :--Of three rectangular axes, X, Y, Z, in space, let one, the Z axis, be taken as the axis of any solid figure then the relation between the area of a section transverse to this, and the distance along the axis, may be expressed by the equation

:

Az

A+BzPC9+ D zr + etc....

(1)

where p, q, r, etc., are any positive indices integral or fractional, and the co-efficients are any positive or negative numbers. The form of

*

the transverse sections is indifferent, that is to say, although 4 is a function of x and y, as well as z, the form of the xy function is not restricted, excepting that it must represent a real area. For example, it may represent any closed curve whatever, and may be so involved with z that the curve changes its form as well as its area as we proceed along the z axis. The relation (1) is thus seen to be of a very general character. The integral of (1) is

This integral will represent the volume of the solid between the transverse section for z=0, and that for the distance z.

2=

Similarly, if A, be an ordinate instead of an area, the integral will represent an area instead of a volume, so that the cases of areas and volumes may be regarded as mathematically identical in respect of the question discussed. Suppose now that the distance 0 to z be taken as unity, the only effect on (1) or (2) will be that the co-efficients will be altered, the degree of the equation will be quite unaffected. Hence in the integral (2) the z terms may then be omitted. Then, if a, b, c represent any proper fractions in say ascending order of magnitude, and α, B, any series of multipliers to be multiplied into the values of A, for distances along the axis equal to those factors of the whole, then ☛ denoting the sum of the multipliers, we may evidently write

В (∞ ao + ßb3 + etc.) + C (a a¶ + ß b¶ + etc.) + etc.

=

(4)

an equation which must hold good term for term.

The paper investigates the relations that subsist between a, a, ß, b, P, q, etc., that is to say, between the position of the sections on the axis, the multipliers that are required for particular sections, and the number of sections that must be taken for a given number of indices. The following propositions, somewhat paraphrased so as to be applicable to the determination of volumes, are deduced ::

(a) If m be the number of indices in the function (1), there must in general be m+1 transverse sections arbitrarily taken along the axis, in order that the ratios of the multipliers shall be determinate.

(b) If the number of indices exceed by k the number of transverse sections arbitrarily taken along the axis, k+1 indices must be conditioned.

(c) If the number of arbitrary transverse sections exceed by k the number of different indices, then k multipliers must be arbitrarily assigned before the remainder can become determinate.

(d) If the original function have only one positive index, a single mean transverse section at a point on the axis dependent upon the index, may be found, the limiting position of which is 0.3678794.

(e) When the original function (1) contains only two indices, one being unity, it is possible to find two transverse sections equidistant from the centre of the axis, and that their half sum is the mean sectional area.

(f) These symmetrically situated transverse sections cannot be at a greater distance from the terminals of the axis than (about) 0-2123179, the axis being unity. At that distance, the indices simultaneously satisfied are 1 and (about) 2:471 only.

(g) If the symmetrical sections be at a distance of 0·1997088 from the terminals of the axis, then the only indices that can be satisfied are 1 and 4.7345.

(h) If two symmetrical sections be at a distance of not more than 0-2123179 from the terminals of the axis, considered as of unit length, then in general the index 1, together with two conjugate indices, the one greater, the other less, than 2.471, can be satisfied.

(i) If the two sections be taken at the distance 0.2113249 from the terminals of the axis, the three indices 1, 2 and 3 will be satisfied.

(j) Two terminal sections and a middle section will in general satisfy the index unity, together with two conjugate indices, the one greater, and the other less, than 2.458: these indices are dependent upon the co-efficient assigned to the middle section, which can in no case be less than 3.933647 (the co-efficient for the terminal sections being unity), viz., its value at the critical index 2.458.

(k) If the co-efficient 4 be assigned to the middle section, the indices satisfied will be 1, 2 and 3, and none other can be satisfied.

(1) When a middle section and two others equidistant therefrom, all with equal weight, are taken, the latter can never be at a greater distance than 0.1469624 from the terminals of the axis, at which distance the indices satisfied are 1 and 2:449.

(m) Sections nearer the terminals of the axis than this limiting value will satisfy the index 1, together with two conjugate indices, the one greater, and the other less, than 2.449. If the distances from the terminals be 0.1464466, the conjugate indices satisfied, together with 1, will be 2 and 3.

(n) When the co-efficients of three sections, viz., two terminal and one intermediate, in a definite position, are so deduced as to satisfy two indices, a third conjugate to these will in general be satisfied, two of the indices, or all three, may, with particular co-efficients, become identical.

(0) When the transverse sections include the terminal sections, are equidistant, and have assigned to them suitable co-efficients, these being identical for sections equidistant from the centre, then, if one of