104 Nev:iton and Phillips on certain Transcendental Curves. 



tiiiuous closed line represents the curve of equation (12) in this 



case. The axis of y is the heavy vertical line, and the axis of 



X the upper heavy horizontalline. These heavy lines are double 



lines. 



h. Several of the propositions of Art. 11 apply to equation (12) with 



p . p' 



evident modifications. If — is not equal to — „ there are no 



straight lines belonging to the locus. 



i. We may regard the plane of the curve as divided into equal 

 rectangles by lines parallel to the axes, the altitudes of the 

 rectangles being qrr, or 25-7?, according as p-\-q is even or odd, 

 and whose bases are q'n, or 25-' tt, according as p' -\-q' is even 

 or odd. The curve (12) repeats itself in each of these rectangles 

 without any variation, through the whole extent of the plane. 



j. The origin of (12) is a real double point. 



18. Effect of a change of the value of a in equation {\),ichenb=iO. 

 The effect of a change in the value of the coefficient of the second 

 member may be observed by comparing some of the figures : for 

 example, figs. 38 with 39 ; figs. 41 with 93 ; figs. 45 with 72 and 73 ; 

 figs. 77 with 80; figs. 43 with 74 and 75 ; figs. 123 with 131-135; 

 figs. 136 with 141 and 145. 



1 9. The effect of the change of this factor can be better observet). 

 in the simpler equation 



sin y= ^ sin x, (13) 



where k represents a as assuming several values. Figure 130 repre- 

 sents a faisceau of curves for equation (13). The origin is the nodal 

 point near the lower left hand corner of the figure. Let k change 

 from — cc to -j- 00 . 



a. If ^'rroo , we have the vertical eqiiidistant straight lines. 



b. If k=. — 2, we have the curved lines represented by uniform fine 



dots. At the origin it is tangent to y= — 2x. 



c. If A:= — 1, we have the straight lines in which dots and strokes 



alternate. 



d. If /<;— — ^, we have the continuous curved lines. 



e. If A'-nO, we have horizontal straight lines. 



f If k=:^, we have the heavy dotted curved lines. 



g. If k=zl, we have straight lines of which y=x is one, and the others 



are similarly marked. 

 A. If k=z2, we have the curved lines consisting of a stroke and three 



dots alternating. 



