102 N^ewtoii CDid Phillips on certain Transcendental Curves. 



continually throughout the plane of coordinates. The curve is sym- 

 metrical about either axis, and also about either line yz=. i.e. These 

 two lines belong to the curve. 



The origin and a portion of tlie curve, principally in the first 

 quadrant, are given in fig. 67, ])late XXII. 



13. Equation (1) when «— : — 1, JirzO, ^/i=in-= — If (j is eiwn the 



equation 



sin y sin — y =: — sm x sin —x (9) 



q q 



merely changes the sign of the second member if we substitute 5';r -fa; 

 for X. Hence the curves in figures 40, 42, 44, and 47 represent equa- 

 tion (9), when q is eium, the origin being at an isolated point. 



But if q is odd we obtain new forms which have these properties. 



a. The origin is an isolated point. 



b. If q=l, the locus consists solely of points (fig. 39). 



c. li q=z3, each point is surrounded by one closed curve (fig. 93). 



d. If q=5, each ^aoint is surrounded by two closed curves (fig. 74). 



e. The resemblance of these figvires to parts of figs. 40, 42, and 44, 



and the law ol' their formation makes it unnecessary to give 

 further examples. 



f. A dot and four suiTounding closed curves in fig. 47, would fairly 



represent the element for equation (9), when ^1=9. 



14. Equation (1) when az=. — 1, 5=0, m:=.n=. ^L. Curves whose 

 equations are of the form 



sin wsini^y=: — sin x sin ^x (10) 



q q 



are shown in figures 69, 71, 99, and 108. There are no straight lines 

 belonging to the locus. The origin is at any one of the isolated 

 points. The first two are placed beside figures 68 and 70 for ease in 

 comparing. 



The following propositions of Art. 11, for equation (7) apply also 

 to equation (10), without change of terms, viz : i, c, c7, g.^ /,J, and h. 



15. Equation (1) when az=.\^ l>=0, m=z], and ji= i_ . The figures 



9' 

 76-79, and 81, represent curves whose equations are 



siny sin2/=sin£«sin-£--a;. (11) 



9' 



