VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 703 



tance between the 1 centres e ; let now the tool be fixed in any place, upon, 

 above, or below the rest ; call mE the distance of the tool from the middle point 

 between the centres (marked e on the rest) d ; then the greater semi-axis of the 

 oval so described will be d + \e, and the lesser semi-axis d — -^e ; and thus 

 both the form and position of the oval will be known. All workmen know the 

 tool must never be raised above the place where it was at first held, and we see 

 the reason ; it would destroy the oval first begun to be turned, and form a new 

 one in a different position. 



But there is another difficulty in turning ovals, especially such as have mould- 

 ings, as picture-frames, &c. The tool generally has all the mouldings formed 

 on it: now if it be laid flat upon the rest, and the engine set to work; the 

 mouldings will in some places cross the plane of the tool (or the top of the rest) 

 at right angles (as in turning circles,) in other places obliquely. This will make 

 the several members of the mouldings leaner or smaller in one part of the work 

 than another. Nor will the case be altered if the mouldings be turned separately. 

 Analogous to this, when an oval is drawn by the trammels, the line described by 

 the pencil will not, as in a circle, be always at right angles to the beam of the 

 trammels. The oval line so drawn will be at right angles to the describing beam 

 only at the extremity of the 2 principal axes where the beam coincides with those 

 axes ; in all other places the oval line and beam make an oblique angle. It may 

 be proper therefore to inquire how much this angle deviates from a right angle. 

 This we shall call the angle of deviation. 



All things as in fig. 1, draw the tangents tm and tn, to the point M in the 

 ellipse and the point n in the circle, corresponding to each other ; then from the 

 nature of the ellipse these tangents will meet each other in the axis CA produced. 

 Draw MG perpendicular to tm, then cms will be the angle of deviation sought. 

 Then the angle mtn, between the tangents to corresponding points in the ellipse 

 and circumscribing circle, is equal to the angle of deviation gms. For because 

 TNG is a right-angled triangle, and np perpendicular to to ; therefore tnp = 

 NCP = MSP, that is, in the triangles mtn and gms, the angles tnm and msg are 

 equal. In like manner, because tmg is a right-angled triangle and mp perpen- 

 dicular to tg, therefore tmp = mgp, and (in the triangle mtn and gms) the 

 angles tmn and mgs are equal ; therefore in the same triangles, the remaining 

 angles mtn and gms are also equal. 



To compute the angle mtn, we have by trigonometry tp^ + pm X pn : tp 

 :: mn : tan. mtn, radius being unity. Call now ca =z t, cb = c, cp = x, pm 

 = ?/, CA — CB (or t — c) =^ d, and we have pn = \/{tt — xx) ; also cd : cb :: 

 PN : PM = \/ {tt — xx) X ^, whence pm X pn = {tt — a.r) X j. Again, cp 



: PN :: pn : pt, whence tp = -— — . Lastly, cd : bd :: pn : mn = v' {n — xx) 



