1349.1 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



363 



the axis of y, the point A will describe a straight line perpendicu- 

 lar to the line passed over by C. 



This is the principle and the arrangement (either with two or 

 with four bars) wliich furnishes the most perfect parallel motion 

 for the piston-rods of reciprocating steam-engines, as indicated by 

 Mr. Scott Russell, in his 'Treatise on the Steam-Engine,' p. 210. 



Preserving the same arrangement, except moving the box B to 

 any other division towards A, to determine the curve traced by B. 

 We have just seen, that under such circumstances, A passes over a 

 straight line; and (since by construction 6 =r e) it is evident that 

 the curve passed over by B, as between the rectangular axes of 

 y and t', will be the same as that which would be passed over by A, 

 referred to the axes of x and y, when we consider c' = b, and b>c. 



In this case, equation 7 becomes, since y = 0, 



2bx" 



x-lb- c) = 2jy\r: .-.x = . - . 



^ ' b—c 



Also, equation 8 becomes, 



(m- + 4x")2 + iy = b^{b + !:)•■, 

 and substituting for x, and reducing, we have — 



This is the equation to an ellipse of which the semi-transverse 

 axis = b-\-c, and the semi-conjugate axis =: 6 — c; the former being 

 coincident with the axis of y, and the latter with that of ,r, as is 

 shown by considering .r, and y, in the equation, as successively =0. 



Now, by substituting v for y', z for a/', and 6' for c, we liave 



the equation to an ellipse whose semi-transverse axis, 64-6', is 

 coincident with the axis vx; and whose semi-conjugate axis, 6 — 6', 

 corresponds with the co-ordinate axis of y. 



From the make of the instrument, 6 is a constant quantity,"and 

 to this jth the sum of the transverse and conjugate axes of any 

 ellipse to be traced in this manner must always be equal; 6' will 

 then be equal to |th the difference of those axes. It is evident, 

 too, that when 6'^ 6, the equation is that of a straight line; and 

 6'= 0, it is that of a circle. 



This little investigation naturally suggests an ellipsograph of 

 very simple construction. It also manifestly points to the deter- 

 mination of the curve described by a point in the connecting-rod 

 of a steam-engine, one end of which moves round in the circle of 

 the crank, while the other end performs a rectilineal, or a circular, 

 reciprocating motion. 



This curve possesses interest by reason of the "very beautiful 

 method of working the valves of the steam-engine, invented by 

 Mr. Melling," and is obviously a kind of oval — oblate at the end 

 nearest the crank, and elongated at the other end. 



Now, using the same notation, and referring to tlie annexed 



figure, in which Oc re- 

 presents the crank-arra 

 =:6; eC, the connecting- 

 rod ^3 c'; and cA, the 

 distance of the given 

 P point from the crank- 



' end of the connecting- 

 rod =c: it is evident, 

 \ ^/ the formula} 7, 8, will 



■^ — " apply, by merely chang- 



ing the sign of c, because the line represented by this letter has 

 now a direction contrary to that hitherto assigned to it. Hence 

 they will become — 



(:!?' + y-') {c' + c)-i- b'^ic' -c) = c'-{c' -c)+2c' [yy" + xx") . 11. 

 and, {c'x"-cxf-^{c'y"-cyY = b-{o'-cY . . 12. 



Thus, suppose the path of tlie piston-rod to coincide with the 

 axis of abscissae x, whilst the connecting-rod attaches the crank- 

 arm and piston-rod together; then y = 0, and equations 11, 12, 

 become, x-{c' + c)-i- b-(c'—c) = c"-{c'—c) + 2c'xx" ; 

 and, {cx"—cxy -\- e'y-= b-{e'—c)-. 



The first of these equations enables us to find ,i", for any value 

 of x; and the second gives the corresponding value of j/". 



Suppose, as in Mr. Melling's arrangement, the connecting-rod to 

 be four times the length of the crank-arm, or d =: 46, and 2c =: e' ; 

 then, at the commencement of the stroke, xz=.b-\- c, and we find 

 ■r" = fc', as it should be. At half-stroke, xz=c'—b; and then, 

 Jr"=3c', as it should be. 



It is only in the case of the crank-arm being equal to the con- 



necting-rod in length, that any part of the curve described by a 

 point in the connecting-rod is an ellipse — and then it could not be 

 during more than half-a-revolution: but the conditions imposed 

 by the mechanism render these proportions inadmissable. In no 

 others is the curve described an ellipse; nor is it correct, that the 

 deviation from a true ellipse becomes more and more as the con- 

 necting-rod becomes shorter. 



I 



LEVELLING STAVES. 



A new Method of Gradunting Levelling Staves, by which they may 

 be much more accurately read and at much greater distances than at 

 present. 



At the present time, when drainage of towns with reference to 

 sanitary arrangements, and drainage of lauds in connection with 

 agricultural improvements, engage so large a portion of public 

 attention, the suggestion of any real improvement in the instru- 

 ments usually employed in practical levelling is certain to be lis- 

 tened to with attention, if not with approval, by the numerous 

 body of professional gentlemen now occupied in conducting the 

 operations alluded to. The great benefit conferred upon engineers 

 and architects as a body by Mr. Gravatt, by the arrangement 

 which superseded the use of the sliding-vane in levelling staves, 

 is too well known and appreciated to require comment: while, 

 however, Mr. Gravatt's method is in general far superior to the 

 old one, there are two particulars in which it is usually admitted 

 to be inferior to it — viz. 1st, the trouble and attention required to 

 read the minute divisions on the staff — an important point when we 

 reflect on the vast number of readings taken in a single day's level- 

 ling; and, 2nd, the difficulty of reading them at any considerable dis- 

 tance. In order to remedy these defects, several eminent engineers 

 and others liave, at different times, proposed methods of graduat- 

 ing, which, however, seem to have failed to supersede that origin- 

 ally introduced by j\Ir. Gravatt. To ascertain the reason of this, 

 as well as their comparative merits in connection with some by 

 other engineers, 1 made a series of experiments on all such level- 

 ling staves, of new construction, as I could meet with; a very 

 brief account of the results of which may not be uninteresting, 

 and should perhaps, in justice to the inventors, accompany these 

 remarks, in which shall be proposed a method of graduating level- 

 ling staves, which will, I believe, entirely get rid of the two dif- 

 ficulties already mentioned, without any counterbalancing dis- 

 advantage. Two levels were used, a 10-in. and a 12-in. focus of 

 Troughton's, in the experiments. The staves compared were as 

 follows: — 



No. 1, Mr. Gravatt's: lOths clearer and more readable at long 

 distances than, perhaps, in any other; lOOths read generally with 

 distinctness at about 8 chns.— 10 chns. should not be exceeded; in 

 this Mr. Williams coincides in his 'Practical Geodesy,' p. 63. On 

 the whole, this staff' has the advantage of all that have appeared 

 since; in which opinion I am supported by Mr. Bourne in his 

 'Principles and Practice of Engineering,' p. 210. 



No. 2, A mode of dividing, invented and adopted by Mr. P. N. 

 Barlow, C.E.; the object being to obtain greater distinctness and 

 less liability to error. The divisions are composed of triangles, each 

 occupying ^ijths of a foot, except that at tlie even tenth, which is 

 diamond-shaped, to render it more defined than in other staves. 

 The chief advantage of this arrangement consists in the greater 

 distinctness with which the point of intersection of the triangle, 

 and the hair-line of the telescope is defined, compared with the 

 horizontal divisions, and their paraUelisra with the hair-line. The 

 peculiar dilRculty of setting two parallel lines to coincide with one 

 another is well known to astronomical observers, who can bisect a 

 dot with greater precision than two of the finest lines are known 

 to agree. — vide Encyc. Metrop. 



No. 3, preferred by Mr. Castle ('Land-Surveying,' p. 255), goes 

 by the name of "Stephenson's," and was first used on tlie London 

 and Birmingham. The lOOths are obtained in the same way as in 

 the common ivory protractor; the lOtlis of a foot through the 

 whole length of the staif are bisected, making the two divisions 

 20ths; and these division lines extend the whole breadth across 

 the staffs. The opposite ends of these lines are connected by dia- 

 gonal lines, each one with its preceding — viz., the left of No. I. 

 with the right of No. II., the right of No. II. with the left of 

 No. III., and so on. And five vertical lines are drawn, at equal 

 distances, along the whole of the stafi^, which thus divides each of 

 these diagonal lines into five equal parts, each being ^th part 

 of wijth, or xtrotli P'"'t of a foot. 



47* 



