[cokek] determination OF THE FORM OF CAMS 55 



point, the same inclination to the circle struck from the origin as the 

 curve has to the axis of x. 



If u, V, be the new rectangular co-ordinates where u = rcos ç, v =■ 

 r sin <p. Then using the transformation 



. u -\- IV = e^ + 'i' — f-^ (^cos y + i sin y) 

 and equating real and imaginary parts we obtain 



r = e^,(f) = y 



This transformation ensures that every straight line x = x^, becomes 

 a circle of radius r = e^ about the new origin, while every straight 

 y =. 6 becomes a radial line passing through the new origin at an angle 

 ^ = 6., or generally lines parallel to the axis of y become transformed 

 into circles round a new origin and lines parallel to the axis of y become 

 radii from the new origin. 



Let now the tracing point z in the x, y plane trace a part of the 

 cam curve such that at any point x, y the inclination is yS, then the 

 inclination of the transformed curve at the point m, u to the circle is yS 

 and hence the required transformation is accomplished. 



The magnification is altered, however, as can be seen at once by 

 taking a line passing through the origin of co-ordinates in the plane icy at 

 an angle a. 



We have y = mx where m = tan a 

 Hence r = ê^/'" , ^ = y 



When y = r = s" = 1 



When y = 7t r = fV"' 



The displacement from o — ;r is therefore g'^/'» — 1 instead oimn 

 By altering the scales it is in general possible to make the displace- 

 ments equal. 



Hence the conformai method of representation enables us to trans- 

 form a curve in the plane of x^ y to polar co-ordinates with the angles 

 unchanged. 



Application of Fourier's Series to Cams. 



It is shown in works on mathematics that any function of x of 

 period 2 n which is finite and continuous can be represented by a series 

 of the form 



Ux) — Ag -\- J^ cos X + A2 cos 2 X -{- A^ cos 3 X -{- 



+ i?i sin X -f- B2 siîi 2x -|- B2 sin Sx -\- 



