1841.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



343 



and is called the pitch of (lie screw ; it could not possibly signify, as 

 understood by Mr. Cussen, the distance from the upper side of one 

 coil of the thread to the under side of the next, as tliat would admit 

 of an infinite luiuiber of different solutions to the problem of finding 

 the power necessary to overcome a given resistance, according as the 

 ratio of the thickness of the thread to the interval between its coils 

 might be more or less, which circumstance could not effect the result, 

 since it is only the upper side of the thread, or that which is in con- 

 tact with the' resistance, wdiich sustains the resistance. Mr. Cussen 

 may, therefore, rest satisfied that aU theorists agree with him in sub- 

 stance, if not in expressions. 



Respecting his second objection, Mr. Cussen has overlooked the 

 chief part of the theory of the /trtr, and, unless he objects to that 

 theory also as now taught by all authors and professors without ex- 

 ception, the following reasoning will convice him of his error. 



We will take his examples of the three screws, each of i inch 

 thread, which, converted into intelligible mechanical language, is one 

 inch pitch, and of 3, 6, and 9 inclies diameter respectively, worked by 

 a lever of HO inches long, the lever moved by a windlass of one ton 

 power. But it is necessary first to understand clearly what is meant 

 by a lever 90 inches long. In mechanical language its signification is 

 the distance in a straight line from the fulcrum (which is in the axis 

 of the screw) to the point of application of the power, which does not, 

 however, seem to be the meaning attached to the expression in Mr. 

 Cussen's second letter ; he seems rather to mean the distance from 

 the siiiface of the cylinder to the point of application of tlie power, 

 which is not the true measure of the power of the lever; we shall 

 therefore take the liberty of understanding it in the former sense. 

 This being premised, suppose for a moment that no lever is used ; we 

 shall have, by Mr. Cussen's, as well as by Mr. Bridges' formula: 

 in the first case 1 ; 3 X 3-1416 : : 1 : ni = 9-4246 tons, 

 in the second case 1 : 6 X 3-1416 : : \ ; w = 1S-S49G tons, 

 in the third case 1 : 9 X 3-1416 : : 1 : w = 28-2744 tons. 

 Now the power is applied in the first case at a distance of lA inch 

 from the fulcrum, in the second at 3 inches, and in the third at Ak in. 

 distance ; so that, by applying it at a distance of 90 inches in all 

 three cases, we shall obtain the following results respectively: 



in the first case 1^ : 90 : : 9-4248 : ?»= 565-4SS tons, 

 in the second case 3 : 00 : : 18-8496 : «) = 565-488 tons, 

 in the third case 4* ; 90 : : 28-2744 : ?» = 565-488 tons, 

 or the ]jressure independent of the diameter of the screw, which over- 

 throws the second objection. 



Mr. Cussen's third objection falls to the ground with the preceding, 

 indeed it has no meaning at all ; for he virtually multiplies by the cir- 

 cumference of the circle described by the extremity of the lever when 

 he multiplies by the circumference of the screw and by the length of 

 the lever, although he omits to divide by the semidiameter of the 

 screw, as he ought in that case to do, and as it will be seen, on an in- 

 spection of the above calculations, we have done to obtain the final 

 value of ic. If we take the first case, we had finally 

 1x3x3-1416X90' 



>»=: ; ■ , 



ixU 



and it is obviously the same thing whether we suppose 3-1416 to be 

 first multiplied by 3, to give the circumference of the screw, and the 

 product to be afterwards multiplied by 90 the length of the lever, and 

 divided by \h the semidiameter of the screw, as above, or whether 

 we suppose 3-1416 to be first multiplied by tioice 90, to give the cir- 

 cumference described by the extremity of the lever, and as the factor 

 3 of the numerator is essentially twice the factor U of the denomina- 

 tor, these two factors disappear. Or, to make it still more apparent, 

 ■^ let r represent the radius of the screw, d its pitch, / the length of the 

 lever (measured from the axis of the screw), P the power and w the 

 resistance. Then the last equation would be 



Px2r X3-1416xi 



n»:= ■ 



dXr 



from which it is evident that, if we take the 2 from the factor 2 r, and 

 multiply it by the two factors 3-1416 and /, we shall obtain the cir- 

 cumference described by the extremity of the lever, or by the power; 

 and this product, multiplied by P X r will obviously be the same as if 

 the product 2rx3-1416, which is the circumference of the screw, 

 were multiplied by P X /. But Sir. Cussen has committed the error 

 of leaving out the factor r in the denominator, forgetting that when no 

 lever is used, the power is applied at the circumference of the screw, 

 and that the leverage is equal to r, so that when tlie leverage is in- 

 creased to I, the resistance is increased in the ratio -. Having demon- 



strated Mr. Cussen's error, and shown its probable origin, we may now 

 cancel the r in the numerator and denominator of the fraction, and it 

 will remain 



PX 277/ 



IT being the ratio of the circumference of a circle to its diameter. 



If Mr. Cussen's remark "that one-third of the calculated power is 

 lost by friction," is meant to bear upon the comparison of the effect 

 of screws of different diameters but the same pitch, it will be found on 

 investigation, that the friction bears no fixed ratio to the resistance, 

 but increases in a slightly greater ratio than the diameter of the screw, 

 and thus gives a proportionate advantage to screws of small diameter. 



ON THE ECONOMY OF FUEL IN LOCOMOTIVES CONSE- 

 QUENT TO EXPANSION AS PRODUCED BY THE 

 COVER OF THE SLIDE VALVE. 



Sir — Having observed several errors in Mr. J. G. Lawrie's calcula- 

 tions, published in your useful and interesting Journal for August last, 

 allow me to point them out for the benefit of your readers. 



I should premise that the formula he has given for the several dis- 

 tances travelled by the ])iston : from the commencement of the stroke 

 to the commencement of expansion, from the commencement of the 

 stroke to the openi-jg of the eduction port (not to the end of expan- 

 sion, for expansion continues, but more rapidly, and the effect during 

 the rest of the stroke is not to be neglected), and from the commence- 

 ment of the stroke to the position of the piston when the valve opens 

 for the lead of the next stroke, are correct. I should however observe 

 that the expressions under the radical sign in the values of a' and c' 

 are identical, and may be reduced to (1 — c') [1 — (^ + c)'']; and per- 

 haps it would be better if the expressions (1 — i c—c") and (1 + / c -f c') 

 in the same two values were written respectively 



[l-e(/-1-e)]and[l-i-c (/ + c) ]. 

 The errors I have discovered are in the computation of the effect, 

 which follow. 



Mr. Lawrie finds the volume of steam of the initial pressure p ad- 



((2 d — h) t\ 

 a' j , (at least 



I suppose this expression to have been meant by the writer, although 

 the factor s is omitted and A is printed instead of p in the denomina- 

 tor,) which is a sufficiently near approximation, but I cannot compre- 

 hend how he can make this quantity equal to 2 (^ X !> although he 

 observes with truth that the quantity of fresh steam must (whatever 

 the expansion is) be constant ; bnt a constant quantity is not neces- 

 sarily an arbitrary one, as which it might be considered in this case, 

 for we may give s any vahie we please, and it would follow that the 

 quantity of steam used per stroke would be the same, whatever the 

 area of the piston might be, provided the length of stroke, lead and 

 cover of the slide were the same. And if we supposed the area of 

 the piston s:=\ square foot (a reasonable hypothesis), the factor 1 in 

 the expression 2 a! X 1 signifying (as I suppose) also 1 square foot, 



we should necessarily have a' — ^ — = 2 «?, which is impossible 



P f 



for a' £ 2 d. His expression of the value of s is therefore incorrect; 

 besides it is obviously impossible to deduce the area of the piston 

 from the length of stroke, cover and lead of the slide, and ratio of the 

 greatest to the least pressure in the cylinder, without knowing how 

 much steam is generated in the boiler. 

 Secondly, the effective working pressure during the expansion is 



found = — — /, .r expressing the distance travelled by the piston 



from the beginning of the stroke ; and this expression will give too 

 great a value by 3 or 4 ft. per square inch, if not more ; for / is used 

 to express the least pressure of the steam in the cylinder, which it has 

 at the moment when the eduction port is closed, and which probably 

 scarcely exceeds the atmospheric pressure, and the mean resistance of 

 the waste steam amounts to 4 or 5 ft. per square inch. Besides this, 

 the formula given to express the quantity of work done during the 

 portion b of the stroke makes no allowance' for the diminution of tem- 

 perature consequent on expansion ; but this may be too slight to be of 

 any consequence, as the expansion is inconsiderable in locomotives; 

 nor is any aUowance made for the waste space which has to be filled 

 with steam. But tJie effect during the rest of the stroke is not to be 



