106 Newton and Phillips on certain Transcendental Curves. 



22. Chayige of value of h in equation {!). It remains to consider 

 the effect of a change in the constant b in equation (I). That it may 

 change entirely the appearance of the locus will be seen by a glance 

 at figures 92, 93, and 94. The same curves are superposed in fig. 95. 

 Though each locus may have its own double points, they cannot 

 when superposed cut each other. 



23. In the figures 96-103, the curves of the equation 



sin y sin fi/= — sm x sin ^x-\-k (16) 



are shown for certain specified values of k. The origin is the place 

 of the isolated point in fig. 99. The several curves if superposed will 

 not intersect. The values of k were selected so as to furnish curves 

 with double points. 



24. A series of twelve curves from the equation 



sin y sin x\y^ —sin x sm^jX-\-k (1 V) 



is given in the figures 104-115. By tracing any selected portions of 

 the figure through the series the effect of the change in k will be 

 seen. As in equation (16) values of k were chosen which give (except 

 fig. 108) real double jDoints. In each case other curves of the series 

 with real double points might have been given. 



25. Another series of fourteen curves is given in figures 116-129 

 from the equation 



sin y sin ^y=sm x s'm^x-{-k. (18) 



The complete series would give 18 curves with double or isolated 

 points. The omitted curves are those having isolated points, one at 

 the beginning and one at the end of the series, one between figs. 127 

 and 128, and one between figs. 129 and 130. 



26. Similar partial series can be seen in figs. 136-138, in figs. 

 139-143, and in figs. 144-146. 



27. The superposition of the several curves of a series is shown in 

 figure 147 where the curves represent the equation 



sin y sin f y= sin x sin fx-\-l: 

 A little more than one complete figure of the curves is shown. The 

 oi-igin is at the double point near the top of the figure. The value of 

 k varies from curve to curve by intervals of -i^, and it cannot numeri- 

 cally exceed 2, The full line corresponds to kz=0. 



The multiple that k is of -j-^ is denoted by the number of dots 

 between the long strokes of the lines. 



The multiple that k is of — i is denoted by the number of short 

 strokes between the long strokes in the lines. 



