Newton and Phillips on certain TransceMilental Curves. Kif) 



i. If X-=-|-a:,we liaA'e vertical lines iigaiii. The curve is at the 

 origin always tangent to yz=ikx. The faisceau has nodal points 

 wherever .r and y are both multiples of n. 



20. If we consider in like manner the faisceau of curves 



sin y sin f y=:A;sin x sin fit', ( 1 4) 



for various values of k (fig. 148), we shall find similar but more com- 

 plicated changes. The origin is the intersection of the heavy lines 

 near the top of the figure. The figure represents the loci for six 

 values of Jc, viz: oc, —1, — |, 0, +1, and -\-2. Each of the six loci 

 passes through each nodal point, if isolated points be counted as 

 branches of a locus. 



a. For kz=. oc, we have the vertical straight lines. The heavy line is 



a double line. 



b. For k=. — 1, we have the uniformly dotted curves. 



c. For k-=i — ^, we have the curves represented by strokes and four 



dots alternating. 



d. For X-=:0, we have horizontal straight lines, the heavy lines being 



double. 



e. For A,=rl, we have the continuous curves (compare fig. 14V). 



f. For A'=2, we have the curved lines consisting of a long stroke and 



a short stroke alternating. 



By removal upward or downward a distance of Stt, the curve (b) 

 coincides with (e). In general any one of the curves by such change 

 coincides with that one for which k has an equal value with opposite 

 sign. 



21. We may in like manner obtain a faisceau of curves from the 

 equation 



sin y sin my-^k sin x sin nx-\-b, (15) 



by giving to k different values. 



The curve will be the horizontal lines siny sin myz=b (Art. 8), if 

 kz=zO. It will be the vertical lines sin x sin mx=0, if kz=0. For 

 other values of k, the curve will pass through all the points of inter- 

 sections of these series of straight lines. Figure 66 represents (with 

 the axes interchanged) the vertical and horizontal lines in a special 

 case. 



The lines of maxima and minima values of x and y, and the pos- 

 sible positions of double points (Art. 11, i, ./, k,), are independent of 

 k and b. The origin is not upon the curve if k and b are finite. 



Trans. Conn. Acad., Vol. III. 14 October, 1875. 



