ISoO.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



133 



centres of rotation — O anil F, slioulcl continue in contact without 

 any other than a rolling; motion one on another, appear to be, tliat 

 if we assume any two points, B and C, such that the arcs AB, AC, 



measured from the original point of contact A along the periphery 

 of each curve, be equal in length: 



1st. Tlie sum of the vectors FC, OB, must be equal to FO; and, 



2nd. That the sum of the angles FCII, OBK, made by the vec- 

 tors with tangents at the points B and C, must be equal to x, 

 or 180°. 

 For unless the first of these conditions be fulfilled, it appears plain, 

 that when, by the motion of the axis at O, the one curve shall have 

 assumed the position represented by the dotted periphery, the 

 point B having been brought to the position B', the point C would 

 not be, as it should be, in contact; and if the second were not 

 fulfilled, the curves would intersect at some other point, instead of 

 having a common tangent at B'. 



I need not take up your valuable space by entering into any 

 detailed proof that these conditions are fulfilled by equal and 

 similar ellipses working on foci, as a very slight acquaintance with 

 the properties of the ellipse is suflicient to show tliat such is the 

 case. That they may not possibly be fulfilled by some other more 

 complex curves, I do not venture to assert, as the problem would 

 be one of such extreme intricacy witli regard to any other than 

 equal, similar, and symmetrical forms; but I do not regard it as 

 probable that any such curves can be found. 



This principle would enable us to obtain motions of rotation of 

 different degrees of variatiuii, but of the same character — viz. with 

 one maximum and one minimum velocity in the course of eacli 

 revolution, according to the excentricity of the ellipses made use 

 of. The revolution of the one wheel is necessarily conterminous 

 with that of the other, but is described at a variable rate; the 

 nature and amount of wliich variation may be readily ascertained, 

 either analytically by means of the formula subjoined, or by the 

 merely mechanical process of drawing an ellipse of the assumed 

 excentricity, and drawing riglit lines from any point on the peri- 

 phery to each of the foci; since it will appear plain, on con- 

 sideration, tliat, for any a-sumed point C, CFA represents the 

 angular motion at F due to the angular motion CGA, or BOA, 

 at O. 



To deduce an analytical formula applicable to the calculation of 

 these angles, we take the polar equation of the ellipse with regard 

 to focus G and origin GA, viz. 



, a- — <r 



a — c . cos^' 

 in which a = semi-major axis, and c=:: the linear excentricity. 



Hence we find that the angle CFA, or cf, representing the angu- 

 lar motion round F, due to the angle CGA, or 0, round O, must be 

 such that 



+ 



= 2a; 



C; — c.cos5 ' a — C.COSf 

 since, in order to fulfil the first of the conditions which we have 

 shown to be required, H or G»i -f Go must be equal to FO or 2a. 



From this we can readily derive, by ordinary algebraical pro- 

 cesses, the expression, 



^ = cos-' ((g!+«')-co sfl + 2a c\ 

 I a^ +c°- +2ac.cose J ' 

 a form easily calculated for any given values of the constants. 

 This action is far more simple, both in theory and practice, than 



that which has been already made use of — elliptical wheels work- 

 ing from tlie centres — the major axis of the one being placed at 

 tlie commencement in tlie same straight line with the minor axis of 

 the other. By combinations of the two, a variable motion of 

 almost any regular periodic character may he attained, by due care 

 in assigning the proportions of the constants; and great facilities 

 thereby afforded for counteracting the effects of any irregularities 

 in the motion of machinery which other circumstances may have 

 induced. 



In many cases it would be possible to economise power and 

 space by such application; and in the hope tliat these hints may 

 prove serviceable to some of your many readers, I have been 

 induced to trouble you with this trifling communication, which 

 you are welcome to deal with in whatever manner may prove 

 convenient. 



Southampton, William Davisox. 



February 20tli, 1S50. 



P.S. Since writing the above, I have seen a small planing ma- 

 chine at Mr. E. P. Smith's engine factory, in this town, to which 

 elliptical wheels, acting in the manner described, had already been 

 applied, with ingenuity and success, to retard the forward motion, 

 and accelerate the return motion, of the cutting tool. I was not 

 aware that the principle had been applied; but as it is certainly 

 far from being generally known, and as it appears to me capable of 

 being applied in many ways with advantage, the publication of the 

 above sketch, thus divested of all pretensions to being the first 

 notice of the principle involved, may still prove useful. 



Southampton, March 12th, 1850. 



THE SMYRNA STEAM FLOUR MILLS 



AND 



THE WATT AND AVOOLF STEAM-ENGINES. 



With reference to an article on these subjects in our last number, 

 we have received a communication from Messrs. Joyce & Co., of 

 Greenwich, which we now insert, and to which we shall append 

 a few observations. It is as follows : — 



TO THE EDITOR OP THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



Sir — In your number for last month, which contains an account 

 of the steam-engines and flour mills recently constructed by us for 

 Smyrna, you have questioned the fact of those engines consuming 

 " less than 31b. of coals per horse-power per hour ; " and add, that you 

 cannot believe such a statement to have been made with our sanc- 

 tion ; we should manifestly be wanting as well in a natural desire 

 to do justice to ourselves as in a proper regard for our professional 

 reputation, did we not avail ourselves of your pressing invitation, 

 or challenge as we may rather call it, to verify or disclaim that 

 statement through tlie medium of your columns. AV'e shall there- 

 fore begin by saying that sucli allegation was made with our entire 

 sanction. So far, however, from its being so extraordinary and 

 unprecedented a performance as to have furnished grounds for 

 your unqualified scepticism, we find you have long since borne tes- 

 timony of having witnessed '■^ the gratifying fact" "'that a rotatory 

 fly-wheel engine for land purposes can be made to do with 3/6. of 

 coal per horse-povver per hour;" for if you wiU turn to your Journal, 

 Vol. v., p. 109, you will find an article emanating from your pen, 

 in which you report a double-cylinder engine constructed by 

 Messrs. Rennie, and erected on the premises of Mr. Thomas Cubitt 

 at Pimlico, to be working at 2^ lb. of Graigola coals per horse 

 power ptr hour ; neither is this "the full indicated power," 

 but the actual duty, you yourself having deducted from the indi- 

 cator diagi-anis an amjile allowance for "friction, the power con- 

 sumed by the pumps, tkc." a deduction which you seem to infer 

 may not ha\e been made in our case. 



As the Smyrna engines could not have been put to work until 

 after their erection at Smyrna, we cannot furnish you with any 

 indicator cards of their performance; but we can, if you think it 

 necessary, after reference to your notice of the Pimlico engine, 

 hand you indicator cards of other engines constructed on this prin- 

 ciple by us, from which you will see that the statement made by 

 the public journals was a very moderate representation of their 

 rate of consumption. 



We do not profess, as you suppose, to have made any new or 

 important discovery in the principle of double-cylinder expansion. 

 All we claim is the simplification of the arrangements by which tlie 

 number of parts, the weight of material, and amount of workman- 



19 



