9. A pipe designed to be 36 in. diameter, and to be laid so as to give a fall of 10 ft. per mile, is required to be enlarged so as to discharge 30 per cent. more water; calculate the diameter, the fall being the same in all. 10. Calculate the cubic contents of the excavation, the lower part being rock, X 12 X 14 X B QX 20 × 5 × X C D A 162 ft. 270 ft. 324 having slopes of to 1, and the upper part clayey gravel, requiring slopes of 2 to 1; the base of the rock excavation at the formation level being 27 ft.; the rock and earth to be given separately, and for each separate block, in cub. yds. 11. Deduce and prove the formula for the cubic content of the blocks of earthwork above the rock excavation below it. 12. The following being the depths of embankment at several successive distances, one chain apart, namely, at A, 18 ft.; at B, 22.50 ft.; at C, 32 ft.; it is required to set out the half widths of the base of the embankment from the centre pegs, the width at top being 30 ft., and the slopes 2 to 1, and the ground on which the embankment stands having a uniform transverse slope of 20 to 1. Calculate the half widths, either as measured on the horizontal or the inclined surface of the ground. 13. Calculate the cubic content of each of these portions of embankment with the data of the last question, namely, from A to B one chain, and from B to C also one chain. 14. In Macneill's Tables for calculating the land required for the base of any line of communication, we find opposite 153 the number 0.0035124. Explain each of these numbers, pointing out how they were obtained, and the method of using them. 15. In the same author's second series of Tables for computing the cubic content of earthwork, we find at the intersection of lines 10 and 21 the numbers-Multiplier 0.5741 Add 4.636. Show how these numbers were obtained, and their method of application. 16. If the content of a cutting be computed by multiplying the length into the middle area, deduce and fully prove the expression for the error in the result, in terms of the end heights and the ratio of the slopes; pointing out from the work whether this method gives a result too large or too small. 17. A girder beam 180 ft. in span is given at the centre a camber of 3 in. above the horizontal line joining the abutments; the curve of this camber being a circular arc. Compute the amount which the platform must be kept up above the horizontal line at the points 20 ft., 40 ft., and 60 ft. distant from either abutment; the values may be calculated either by the exact formula, or by the approximation for such ordinates given in Cotton's Manual. 18. Describe the method of manufacture and constituents of Portland cement, and the cause of the failures of former efforts to produce it from these same materials; and give the details of the method of testing each cask. 19. State the qualities of the Portland in contrast with the Roman cement in their application to engineering structures, describing the method of manufacture and chief source of the material of the latter. Put down a contract price of brickwork set in lime mortar, and in Roman cement per cube yard, respectively. 20. Deduce the expense of hauling earth from an excavation per cubic yard per mile, by horse power, stating the average tractive force, duration of day's work, and best rate of travelling for this mode of transport; the line being supposed level, the friction at Toth, and the weight of the waggons one-fourth of the gross weight drawn, assuming some particular rate per day for the horse hire. 21. Day, in his work on Railways, gives an average estimate of the number of cube yards per day per man that may be turned out of an excavation, and hence, at a stated rate of wages, deduces the cost per cubic yard of this part of work. Add also his rule for, and estimate of the cost of tipping. 22. State the Regulations of the Board of Trade which must be complied with for their inspecting officer previous to the opening of a line of railway, as to the stations, platforms, signals, points, and crossings, &c. 23. Give the specifications and general dimensions of a weir when constructed upon the bed of a river consisting of rock, and in part of gravelly clay; and describe the peculiarity of the Shannon at Killaloe as to its plan, and the modification adopted in the weir at that place in reference to this. (It is not required in the answer to this question to include any coffer-dam that may be required at the works.) 24. Enumerate and specify the several descriptions of stone masonry employed in the construction of railway bridges. 25. An experimental wrought iron beam 20 ft. in clear span, and 16 in. deep at the centre, having 4.30 sq. in. in the top, and 2.40 sq. in. in the bottom flange, and 1.90 sq. in. in the vertical web, is proposed to be loaded with one-fourth of its ultimate breaking weight; compute this weight by the principle of the bent lever. 26. Neglecting the vertical web, calculate the position of the neutral axis of the beam in No. 25, and deduce the formula given in Clark on the Britannia Bridge for the moment of inertia of the central transverse section; the distance from centre to centre of top and bottom flanges being taken as above, 16 in. 27. A cast iron beam, 28 ft. clear span, and 2 ft. in extreme depth at the centre, is intended to carry 20 tons uniformly distributed; design the central transverse section so that all parts may be properly proportioned, and the greatest tensile strain not exceeding 1 tons per square inch; stating the formula by which the design is regulated, and the experiments by which it was established. 28. From the central section in No. 27, calculate that which would be sufficient at points 4 ft. and 8 ft. on either side from the centre, the depths at those points being 1 ft. 11 in. and I ft. 8 in. respectively. 29. State the chief dimensions of the Grosvenor Bridge at Chester, the alterations which were made in the design from what occurred on open ι ing the foundations, and the precautions adopted in setting the voussoirs so that the pressure might be uniformly distributed on the beds; and, lastly, describe the method of setting the key course. 30. Give in answer the same details as in No. 29, for the bridge over the Dora Repaira at Turin. 31. Describe and sketch the mode of construction and chief dimensions of the shaft erected by Brunel for the southern entrance of the Thames Tunnel, and give a description of the method of sinking it into place. 32. From Simms on Tunnelling, give the description of the method of getting in the shaft lengths and side lengths, as to the excavating, timbering, and brickwork. 33. A river whose surface inclination is at the uniform rate of 1 in 550 is raised 4.32 ft. by the construction of a weir; calculate by the approximate method the distance up stream at which the surface of the river will begin to be influenced, and also the vertical rise at the point 12 chains distant from the weir. 34. A district 6.3 square miles in area discharges into an impounding reservoir, at which there is a regulating overfall weir 100 ft. long on the crest; a maximum rainfall of 2 inches in 24 hours has been recorded; and this being delivered off the land at a uniform rate in the same interval of time, and the reservoir full at the commencement of the rainfall, calculate the depth at which the water flows over the crest. Examiners. ANDREW SEARLE HART, LL. D., Vice-Provost. HUMPHREY LLOYD, D. D. JAMES HENTHORN TODD, D. D., Regius Professor of Hebrew. K= CHARLES GRAVES, D. D. THOMAS STACK, M. A., Regius Professor of Greek. FELLOWSHIP EXAMINATION. K' I -k sin am 2 Ko K' π 2 Ko I + k sin am 2 Ko π 1. Express sin am by a fraction whose numerator and denominator consist each of a series of infinite products, and determine the value of the constant multiplier. 2. Prove that = do " I 1-k2 sin 2p I - k2 sin 2pdo]. k2 + k22 = 1, 2n-1 II n=∞ I-29 2 sin σ + q2n-1 ̧ 2n-1 n=1 2 Π II denoting the product of all the terms formed by attributing to n all integer values from unity to infinity, and hence derive the development 2 Ko of sin am and cosines of multiples of σ. 2 Ko cos am in series proceeding respectively by the sines " π π 3. Prove that (1+xz) (1+x3z) (1+x3z) X *(1+) (+) (+) I+ and deduce from this formula the values of N=∞ II (1 − 292n-1 COS 20+ g4n-2), nal Ꮎ 2=0 (1-292" cos 20+ gån sin σ II sin σ II expressed in series proceeding by the sines and cosines of multiples of σ. 4. Find the value of π 5. Sum the series = [ I - sin am 6. Prove that tan-192 I sin σ and deduce thence the value of → (0), where 2 Ko 2 Ko π I + } @2 = 92 + } 93 9and deduce thence the sum of the series when σ = · 29 cos 20 +291 cos 40 - 29° cos 60+.... tan-1 qa — tan-1 qa + tan-'q — . π + 3 tan1 qa2o + 5 tan ̄1 qa+..... = 2 K'S Kkdk + KS K'k'dk' : 7. Prove that Legendre's relation between complete functions with complementary moduli can be presented under the form 81 8n-2 80 т == 1 8. Transform the expression K {(1 + k22) K − 2E} by Lagrange's scale of decreasing moduli. 9. Prove that Sn-2 Sn-1 82n-4 I Kdk (1 − q) (1 − q2) (1 − q3) . . . = 1 − q− q2 + q5 + q7 − q12 − gl5+... and determine the law of the indices in this development. 10. If s, denotes the sum of the rth powers of the roots of an algebraic equation, show how the coefficient of the last term but one of the equation of the squares of the differences can be calculated when the value of the determinant |