[shaw-henry] simple PERIODIC CURVES 143 



In finding a "parent curve," that is to say, one wHich will project 

 into a given one, the problem arises to determine what angles of d 

 and are required, and what magnitudes of intercepts on x and y will 

 produce the given values on x' and y'. 



If A and B are intercepts on the x and y axes, which project into 

 a and h on the x' and y" axes respectively, then it can be shown that 

 the required projection constants are given by 



6 = 01 where Oi is the solution of 



b^cos^ e • cos2^ - (r^a^ + b^) cos 6 + r-a^ = 



(j) = cos"^ (b cos e/ra cos 6) 



A = a/cos 6 



B = rA, where r is the ratio between B and A as they are drawn, 

 and includes the effect of any difference in scale units between the 

 abscissa and the ordinate. It is thus necessary only to measure a, 

 h and e on the given graph, and to adjust r, to be able to construct a 

 parent curve, which will project into the given one, provided that the 

 character of the parent curve is known. In this way many "trial and 

 error" tests may be avoided. In the case of the estuary tidal curves, 

 it will be seen that the ordinary sine or cosine curves might be ex- 

 pected to give, upon suitable projection, fair approximations, at 

 least if the tides are considered one at a time. It will also be apparent 

 that a series of curves of similar design but with amplitudes and axial 

 inclinations which vary in a systematic manner, should for certain 

 classes of similarity, be projectable as a whole, one into the other. 



In Fig. 1, two curves are shown which are each a projection of 

 the graph of x = A cos py, where p = 7r/2B = 7r/2rA, A and B being the 

 semi-axes of each loop, in this case the intercepts on the axes of 

 reference (x is vertical, y is horizontal). The vertical semi-axis, or 

 amplitude, A = 4 cm., and r= -79. 



Curve (a), in which the inclined amplitude a = 2-82 cm. (equal 

 to A cos 6), is a sample projection when ^ = 45° and 0= 14°45'. (It 

 should be remembered that the inclined axis of each loop is parallel 

 to the x' axis, and that the horizontal axis is the y" axis which is the 

 projection of the y axis on the initial plane, and which makes the 

 angle e with the y' axis.) 



Curve {h) in which a = 3-91 cm. is a projection of the same graph, 

 with (9 = 13° and = 60°. 



In order to test the usefulness of the formulae for predetermining 

 6, (f> and A when the projection is given, and also to test the sensitive- 

 ness of the apparatus, the two curves {a) and {b) with others, were 

 measured, and in each case the values of 6 and which had been used 



