[shaw-henry] simple PERIODIC CURVES 145 



Before proceeding with an account of the experimental tests on 

 tidal records, it is perhaps desirable to submit a list of a few element- 

 ary relations, which may save the time of any reader desiring to con- 

 sider the scope of the method, or to check any actual cases. 



List of Useful Direction Cosines 



{t! is the direction of projection) 



For X axis with reference to x' y' z': li = cos d, mi = o, ni = sin Q. 



For y axis with reference to x' y' z' : \%= —sin ^.sin 0, m2 = cos 0, 

 n2 = cos 0.sin 0. 



For z axis with reference to x' y' z': l3= —sin ^.cos 0, m3= — 

 sin 0, 'n3 = cos ^.cos 0. 



For x" axis with reference to x' y' z': cos e = TtXtl^{\-x(), sin e = 



For y" axis with reference to x' y' z': —sin ^ = \i\ s/ {X'Xii) , cos e = 

 ms/VCl-n^iO. 



For X axis with reference to x" y" z":Li = V[(l-nD/(l-nDL Mi = 

 -n]n2/v'(l-n3, Ni = ni. 



For y axis with reference to x" y" z" : L2 = 0, M2= VCl-n'), N2 = 

 n2. 



For z axis with reference to x" y" z": hz = -nily/{\-n2, M3 = -n2Li, 

 N3 = n3. 



These and other direction cosines of possible interest in a quanti- 

 tive examination of the method, can be deduced at once from the 

 properties of direction cosines, and the definitions of 6, and f. 

 It is useful to note that cos e = cos 0/cos0' = cos ^'lcosd = m.il \/{\-n^ 

 where '■I' is the angle between the x and x" axes; also that sin 0' = n2. 



§3. The Application of the Method to Estuary Tidal Curves. In 

 Fig. 2, curve (a) shows a tidal curve for Nelson, Hudson's Bay, copied 

 from the tide-gauge record. Curve {h) shows the form of this curve 

 when projected with ^ = 54° and = 27°, and the superimposed broken 

 line shows part of a pure sinusoid plotted with r= 2-52. 



It will be seen that the possibility of expressing a wave length of 

 this curve in terms of a sinusoid and three constants is clearly demon- 

 strated. Several other samples were tested, and in each case an 

 approximate projection of this kind was obtained. Although this is a 

 quicker way to handle the curves experimentally, it will be seen that 

 the converse proqess to which the previous notes refer is preferable, 

 because it is desirable to produce the tidal graph in the same form that 

 it assumes on the gauge. 



It has been pointed out that the chief application of this method 

 to tidal prediction probably lies in the case of the estuary tides. If 



