Neioton and Phillips on certain 'Iranscendental Curves. 101 



e. Straight linos itanvllel to y=i^.r and cutting tlu' axi's at intervals 

 oiqn, or 2fy7r, according as ]>-\-q is even or odd, belong to the 

 locus of equation (7). 



f. Tliese sti-aight lines divide the plane of coordinates into ecjual 



squares for any given value of m. Each square contains a 

 similar and equal portion of tlie locus, though not always 

 siniilarly placed, 



g. Eqixation (7) is satisfied if sin.v sin?y^T=iO, and siny siii///y— 0. 



Hence the locus passes through all the angular points of the 

 rectangles formed by these two series of parallel straight lines 

 (Art. 7). 



h. If 2^-\-Q is ^<^<^ isolated points appear, belonging to the locus, at 

 the centers of the squares. 



i. The maxima and minima values of y are determined by the equa- 

 tion £- tan a*:^ — tan^^a? (Art. 4, c). This equation represents 

 q q 



straight lines parallel to the axis of y. There are 2(jij) + (/) of 



the lines (Art. 4, h) in an interval of 2q7T. 

 j. The same equation in y gives the maxima and minima values of x. 

 k. These equations are also the conditions of the isolated and double 



points. Hence there can be isolated or double points only 



at the intersections of the lines i- tan xz= — tan ^-x with the 



q q 



lines -L- tany=: — tan i-y. 

 q q 



I. The propositions {i), {J), and (k) hold equally true for any values 

 of a and b in equation (1), and there are similar properties if m 

 is not equal to n. 



771. The figs. 48-65, 68, and 70, represent curves belonging to equa- 

 tion (7). Any point where two straight lines meet, and that is 

 surrounded by an oval may be the origin. 



91. Tf through the double points on the line y=:x vertical and horizon- 

 tal lines be di'awn, these lines will pass through all the points 

 of maxima and minima ordinates and abscissas. By their 

 intersections they will mark all the possible positions of double 

 points for any values of a and b. 

 12. Equatio7i (1) lohen a=l, b=iO, m-=.u=:an irrational number. 



The equation 



sin y sin \/i y=sin x sin \/i ;r, (8) 



represents a class of curves tliat do not repeat their forms but change 



