Newton ant/ JViiUips on. certain Transcendental Curuefi. !();< 



Ill tlu' tlirection of// tliev repeat at intervals of tt. In (ln' direction 

 of .(• they repeat at intervals of qTT^ or 2(/'7r, according as p'-\-q' is 

 even or odd. 



Fig. 80 gives a similar cnrve except that r/= — I. 



16. Equation (1) idieji a=^\, b^=.0^ in-z l^^-Mu\n,=:i-L. The eqna- 



q q 



tion (11) is a special case of the equation 



sin y sin ±- y := sin a; sin :^ a-. (12) 



q q' 



Examples of curves from equation (12) are given in figures 82-91, 

 123, and 141. The number of different curves that this equation gives 

 us is quite large, even if q and q are limited to small numbers. If 

 11 is the maximum value of q and q' , the number of inde]>endent 

 curves belonging to the equation is nearly a thousand. Equations (5), 

 (6), (7) and (11) are special cases of (12). 



IV. Further consideration of the curves of equation (12). 

 a. If the parallel straight lines sin,r sin:^ir=0 be drawn (Art. 7) the 



plane of coordinates is divided by those lines into portions. 

 When two lines coincide the portion between them may be 

 regarded as real but infinitessimal. In crossing any of these 

 lines the sign of the second member of (12) changes from plus 

 to minus, or vice-versa. 



h. In like manner in crossing any of the parallel lines sin y sin ^y=zO, 



^ q 



the sign ot the first member changes. 



c. The lines dny sin ^(/--O, and sinic sin^,r=0, divide the plane 



q q 



into rectangles (some of Avhieh are infinitessimal). The curve 

 of equation (12) passes through each of the angular points of 

 these rectangles. 



d. Since the signs of the two members of (12) must be alike the curve 



passes at any angle of a rectangle into the rectangle vertically 

 opposite. It passes from a rectangle only at the angles. 



e. If, however, any rectangle is of infinitessimal breadth and finite 



length, the curve at its extremity becomes tangent to the line 



that limits the infinitessimal parallelogram. 

 /. If a rectangle becomes infinitessimal in both directions, the curve 



has at that point an isolated or a double point. 

 f/. The horizontal and vertical lines of fig. 148, and the rectangles 



formed by them, illustrate the above propositions. The con- 



