230 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[July, 



So a 



= cse s. sec - 

 fa r 



whence 



V =: !'. cse « 



■I "*" 2r) 



111-, observing that - ^ ;', and passing to coninion logarithmic tables, 

 r 



V = nep. log iO. )• cse s. log tan (-15° + ^ ) 



whence by inversion 



1 * ,A c-j 1 ' \ M. sin s 

 log tan (45^ + o /~ ' ^' 



from which the values of i can be very easily found ; especially when 

 tliey correspond to equi-dittbrent vajnes of r. 



The expert computer will now perceive at a glance, that all the 

 operations needed to determine the co-ordinates of the various points 

 may now be arranged in a simple tabular form so as to require scarcely 

 any figuring. 



I now proceed to the Elliptic Oblique Arch. Put r for the hori- 

 zontal and p for the vertical radius ; the equation of the curve then 

 becomes 





1 



which takes the place of (B). 



This equation may also be put under the form 

 !( := r sin c, - ^ P cos a. 

 where a is the inclination of the trammel bar that would trace out the 

 ellipse ; from this we find 



5 ;) cse s ( , ., „ , ,. 1 



;— == I 0- — p-; cos a -4- p-.sec a ( whence 



tt a r ^ ) 



V = \ (r' — p") sin a -j- p' nep log tan (45 + | ) } 



we obtain 



V cse s ( , _ . )■ ■> 



otherwise we obtain 

 8 



V = cse s 



{^- 



At first glance it might be thought that this equation gives a new 

 curve; it is, however, still a double logarithmic, having its parts de- 

 termined in the manner already described. 



To find the side elevation we have 



5 - ^ — )• p - \/ir - z' 



— =^ tan s 



Sr p^ + (,-_p^)~ 



whence 



. = c.u{^,^+f,.„^(^±^)). 



it is, however, more easily determined thus 



,■- _|- p- tan a 



= cot s 



r p tan a 



=: cot I 



( >' P 1 



l - cot n -j tan a , 



(p i- 1 



But I z ^ — p sin B. 5 o. 



— cot « 



) r 

 I 



— sin 



a + - nep. 1. tan (^45 + ° ^ ] 



For the end elevation we have recourse to equation (P) which gives 



-^ = — / sec a — cos a I and thus 



5a r I J 



P- 



2/ == ; / nep. log tan (45 -|- i a) — sin a i 



■which is the equation of the tractory modified by the existence of the 



•factor -. From this equation the determination of the individual 

 r 



point is most easily obtained. 



I now proceed to consider the Parabolic Arch. _/ being the focal 

 distance, the equation of the parabola is 



■«- = 4/ --, whence a d u — 2/d ' 



whence again the equation 



r = cses.l „ + -^, J- 



which belongs to the horizontal projection; also 



n' 1 



,r = cots.|v/-- { 3-|-_J- } = 

 and also 



X r= cot s 



.■ { 2 « + -^ , or 



y- = 



4 z' 



97 



which are the equations of the three projections. 



I have now run over the equations which serve to determine the 

 difl'ereiit parts of oblique, circular, elliptic, and parabolic arches, and 

 had intended to supply examples of the requisite calculations; but 

 after proceeding to some length in this, it occurred to me that those 

 who have followed the preceding investigations stand in no need of 

 such illustrations, and that these, therefore, would merely occupy room 

 without being productive of any benefit. 



HARBOURS (SOUTH EASTERN COAST.) 



./ Copt/ of lilt Report of the Commissioners appointed to Surrey llie llnrhours 

 of the South-Eastern Coast, to the Lorda Commissioners of the Admiralty^ 



WITH AX P;N'GBAV1N'G^ plate XII. 



Having completed tlie in<|uiry on the subject of tlie H.irl)ours on the 

 Soutli-Easteni coast of England, -wc request you wiU lay l>efore the Lords 

 Commissioners of the Admiralty the result of our investigation. 



Mr. Wood's letter of the 'i.'jth of July last conveyed to us the directions of 

 their Lordships " to -^isit tlic coast between the mouth of the Thames and 

 Selsea Bill, and to examine and report on the state of the existing liarhmirs 

 between those points, -with reference to tlieir being available as places of 

 shelter for vessels passing through the channel, in ease of distress from 

 weather, and also as places of refuge for merchant vessels fioui enemy's 

 criuzers in time of war, and more esjiecially as to their being made stations 

 for armed steam-vessels emi»loyed for the protection of our trade in the nar- 

 row part of the channel;" for fthich ])urpose, tln^ harbours being acccssihle 

 at all times iif tide, and their capability of defence, were stated to be most 

 important considerations. 



Their Lordships furtl er desired us " to report as to what situations we 

 would recommend as best calculated for these various purposes ; whether in 

 any of tlie existing harliouis, or at any other places within the assigned 

 limits ; and also what works would lie necessary to render them available ; 

 and what the probable expense of the undeitaUing would be." 



Before entering into the details of the suhjcct, it will be proper to state 

 that a question arose whether it fell within the ])rovinee of the Committee 

 to offer any remarks on those harhours which were found on inspection to be 

 incapable of access at all times of tide. 



A perfect harbour of refuge, we understand to mean, such as is capable of 

 receiving any class of vessels, under all circumstances of wind and tide. 



Now there is no such harbour along the whole range of coast from the 

 Nore to Selsea Bill ; nor are any of the existing harbours capable, by any 

 improvements or alterations to their inesent entrances, of being made <aeeesi- 

 hle at low water even to the extent of six feet, with floating berthage inside. 



Most of the harbours on this part of tlie coast arc formed by piers carried 

 out from the main land, and are tidal harbours, dry or nearly so at low water, 

 with l)ars at their entrances ; these liarbours would therefore be excluded 

 from our consideration, if their capability of being made available at all times 

 of tide was to be considered a neeessaiy condition. 



There can he no doubt, however, that the existing harbours are of import- 

 ance to iiiercbant vessels of the smaller classes .at various times of tiile, ac- 

 cording to their dr.aught of water; and though they may not be capable of 

 receiving a large ship, may afford shelter to a smaller one ; and therein' be- 

 come a harbour of refuge to a class of vessels the most numerous and least 

 prepared for heavy weather, or to cscajic an enemy in time of war. 



The value of such imperfect harhours is also increased by the diminution 

 of late years in the size of trading vessels. The large class of sliips which 

 were emjiloyed in the M'est India, and the still larger in the East India trades, 

 have been Bueceeded by vessels of much smaller tonnage. The coasting and 

 coal trades are carried on in vessels of comparatively light draught of water ; 

 and steam-vessels, whose draught is easy compared with sailing-vessels of 

 eipial tonnage, arc rapidly increasing in number, and often supply the places 

 of the larger class of vessels which were formerly employed in the merchant 

 scryice. 



