NeuitoH and PhUUpn on certain IVanscendental Curves. 00 



obtain by iiispoctit)ii all the values of x to a sutHcieiit degree oi" 

 accuracy. 



7. Equation (1) when a=zb=zO. The ecjuatiou 



siny sin wymO, ['A) 



consists of the two equations sinyzrO, and sin ?>iy=:0, and is satisfied 

 by the values i/:=l7r, and i/ii/=f7r, where / is 0, or any integer. In 

 fig. 60 the horizontal lines belong to the equation sin // sin |- ;y^=0. They 

 consist of two series, one at intervals of ;r, the other at intervals of 

 2^7r. Tf through the intersections of the curve in fig. 25 with the 

 axis of .(• there be drawn lines peri)endicular to that axis, the lines for 

 smx sin |a*r=:0 would be obtained. The heavy lines of fig. 60 I'epre- 

 sent double lines, corresponding to points of tangency in fig. 25. 



8. Equation (1) ivhere a=0. The equation (1) becomes by mak- 

 ing a=0, and for convenience changing the axes, 



sina'sinwia-^J. (4) 



This does not contain y, and therefore represents straight lines 

 parallel to the axis of y. If the straight line y=b be drawn parallel 

 to the axis of a^ to cut the curve y=s\nx sminx, and through the 

 several points of intersection straight lines be drawn parallel to the 

 axis of y, these lines will evidently be those represented by the equa- 

 tion since iimutxz=.b. 



In fig. 60 the vertical lines rei»resent the equation sinx sm^x:='f. 

 If the curve in fig. 26 be cut by a line parallel to the axis of x and 

 distant from it two-fifths of the largest ordinate, the intersections will 

 correspond with the intersections of any horizontal line in fig. 66 by 

 the several vertical lines.* 



9. Equation (1) iohere a=\, b=0, m=zn—\. The equation 



sin y sin ?/=sin x sin x (5) 



becomes sin y= ±sin .'■, or ;/z=Itt ±:-'', I being 0, or an integer. The 

 cvirve consists of two series of parallel equidistant straight lines, the 

 one parallel to //=.>', the other to y=-x, and both cutting the axes 

 at intervals of rr. The locus is represented in fig. 38, where the origin 

 is any point of intersection. 



1 



10. Equation (1) v^here a=\, b=0, m.=n= — . The equation 



sin y sin— V— sina; sin —x (6) 



q' q 



is one of the simpler examples of equation (1). 



* The unit of abscissas in plates XIV and XV is smaller than in the other plates. 



