194 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



June, 



against the vessel — anci above all, against the paddle wheels. The 

 ordinary strains of tension arc, hesidos, continually varying ; and these 

 causes combined continually tend to loosen and weaken the several 

 parts of the machinery, and ultimately to produce disrupture. The 

 evident and only way to lessen the probability of such accidents is to 

 make the parts most liable to strains as massive as possible, and to 

 avoid all unnecessary gearing. But the most serious objection which 

 the author urges against the present forms of marine engines is the 

 alleged enormous absorption of power by them, and consequent loss 

 of useful effect. The mrthod he adopts to estimate this loss is con- 

 fessedly merely an approximation — and, as we hope presently to show, 

 an approximation which, being based on unsound principles, is alto- 

 gether wide of the truth. The resistance of water to a moving sur- 

 Tace having previously been determined by experiment fur a given 

 velocity — and resistance being assumed to vary as the square of the 

 velocity — it is clear that if we know the rate of the vessel's motion, 

 and also the rate of motion and the diameter of the paddle wheels, 

 and the depth to which they are immersed, we can cah-ulate the 

 "work done" during a given time. As the subject is one of eonsider- 

 able importance to engineers, we shall proceed to investigate formulae 

 for the amount of work done, and useful effect produced by the rota- 

 tion of a single piddle board. We shall suppose the paddle boards 

 rectangular and perpendicular to the edge of the wheel, so as to radiate 

 from its centre. Let2o=: / of thi' wheel immersed; w =z angular 

 •velocity of the wheel; A= length, 6 = breadth, of paddle boards. 

 Lety"= resistance against an unit of surface for a velocity 1 ; f = ve- 

 locity of vessel; flr= angle passed over from the vertical by the given 

 board at time /. Then the resistance against a thin slip of the board, 

 at distance r from the centre, will be bf (ar — v cos 6)" 5 r ; and the 

 work done while it passes through a very small angle 5 e will be 

 bf(ar — vcos ey r ieS r; also the useful effect — that is, the por- 

 tion of the work done that propels the vessel forwards, will be 

 b/(<u r — V cos 0)- r cos e S r S e. Therefore, integrating bi'tween 

 proper limits, the total work done by one paddle board for one revolu- 

 tion of the wheel is 



b/,[^-^{(l+hr-l*}— ^- sin a. V {(I +hy-P} 



ri^ + ^L-Jl}], 



and the useful effect — 



, , sin 2 o , ,. 



6/ {|{ 0,- sin a {(l + hy- i-} - ^^. ra {(l+hy-P) 



w V 



2e' 



J+Vy-F] 

 2 J 



— g- sin 2 a { (l+hy—l' } 4- (2 D- sin a - ~ (sin a) 

 If h be very small compared with I, these expressions become 

 b/lhf2a o,* P—iu I sin a. « -I- «=(«+ i sin 2 a |' 



and b/ 1 A ' 2 <•>'• I- sin o — 2 I a v a— I a v sin 2 « 



+ 2 f'sino o^C^'" »)' f 



The expression, which Mr. Gustafsson obtains by an empirical and 

 not very intelligible method, is equivalent to — 



2h. ( . 2h^ sin 



-_}„_^__ 



sin a 



l>/hil+' 



i ( 5 



• ' X 2 a for the work done. 



When A is small, this becomes b 1/ h (l w — » I ' x 2 a. 



This subtracted from the correct formula, gives 



bflh V" {a-\ ~ — 2- • > ; which deficiency will ac- 



2 a > 



count for, at least, some of the power Mr. Gustafsson asserts to be 

 lost in marine engines. 



If o = - the above difference = 6 i/A r'{-—- =6Z/A»'X'i 



nearly ; a large quantity, when we consider that in most of the ex- 



periments on board the Acheron I a ■ 



-would, probably 



fof a = -, be less than -, and therefore, the result obtained less than 



half what it ought to be. 



IL Some credit, however, is due to the author for ins ingenuity aud 



perseverance in availing himself of the means of experiment within 

 his power. That more power is expended than useful effect obtained, 

 in sea engines, no doubt is true— but the loss h not so much in the 

 machinery itself, as in the mode of projiulsion. If the floats be radial, 

 the paddle wheels deeply immersed, and the velocity of the boat nearly- 

 equal to the relative linear velocity of the floats, — the resistance 

 against the floats will be most powerful when their useful effect is 

 least, and the greater jiart of the ]iower will be lust. To remedy this, 

 has been the subject of frequent patents; — various contrivances have 

 been proposed to cause the floats to enter the water always vertically, 

 and to preserve as much as possible their vertical position when in the 

 water. But we believe the complexity of the machinery necessary to 

 effect this kind of action, has hitherto been a bar to its adoption. As 

 to the merits of Mr. Gustafsson's proposed form of an oscillating en- 

 gine — and his new method of feeding the furnace — we leave our 

 practical readers to judge for themselves — we having already over- 

 stepped the limits we had assigned to the present review. 



A nhon Treatise on Ihe Steam Engine. By James Hann-, A.I.C.E., 



Mathematical Master of King's College School. London: John 



Weale, 1847. Part I., 8vo., pp. 102. 



Mr. Hann's mathematical abilities are so well known, that his name 

 alone is sufficient recommendation to any work he mav publish. He 

 has evidently presumed that his readers are sufficiently well ac- 

 quainted with mechanical principles, and especially with the theory 

 of the steam engine, to render superfluous any explanation of techni- 

 cal terms connected with those subjects. In fact, the work before us 

 is rather a class-book of reference than an elementary treatise — and 

 as such in a compendious form, comprises nearly all the facts that can 

 be arrived at by purely mathematical investigation, with reference to 

 the application of steam power. 



We wish, however, that Mr. Hann had beeft more rigid in his de- 

 monstrations, and had based them more directly on the dynamical 

 equations of motion ; for instance, in finding the relation between the 

 pressure of the steam admitted, the load, the length of the stroke, 

 and part of the stroke when the steam is cut off — he assumes "the 

 work of the steam must be equal to the work done upon the load," 

 and apparently as though it were axiomatic, instead of being a pro- 

 position capable of a proof, vfhich is very short, and as follows : — 



Let X be the pressure of the steam when the load and piston has 

 been raised through a space .r; let ?«= mass of load and piston; v 

 the velocity of piston; L the load ; — t\ienmvdv=:X dx-Ldx: 



Let 1=^ length of stroke; then, when 



x = 0, and x = l, v=io. 



.•.0 = / Xdz—/ hdx; 

 .•.o = /"'xdx-hl; 



I Xdx\& the work of the steam, and L / is the work done on 



the load : .• . the work of the steam is equal to the work done on the 

 load. The way in which Mr. Hann has stated this relation, would 

 lead one to suppose that it were true for any portion of the stroke — 

 which is not the case. Again, in finding the velocity v, which we 

 have obtained above, Mr. Hann most unnecessarily makes it depend 

 on vis vira and work done — using, not only in this instance, but 

 throughout the course of his book, the former of these two terms as 

 though it implied something more than a merely analytical expres- 

 sion — as though, in fact, it were some independent property of force 

 — " living force," in short, as he translates it. The term via i-iva, as 

 used by modern mathematicians, is a purely arbitrary and conventional 

 expression for certain algebraical symbols. The old philosophers, 

 however, used it to express some confused or mistaken notion re- 

 specting inherent properties of matter. It seems injudicious (to say 

 the least of it) to revive, by translation of the phrase, exploded ideas 

 which were only excusable in an immature state of science. 



" On the Work done by the Engine on the Piston per minute. 

 " Let E represent number of cubic feet of water converted into steam pe 

 minute, A the area of the piston iu square feet, / = actual length of stroke 

 a = that part of stroke before the steaai is cut off, P = pressure in th 

 boiler, jii,= pressure iu the cylinder before expansion, /; = pressure at the x"» 

 foot of the stroke, c = total clearance, N = number of single strokes per 

 minute, V,= work of steam on piston per minute. 



but 



