100 Nev)ton (Did Phillips on certain Tnuiscendental Curves. 



a. It is satisfied if y^=-'X.^ or if y-=. — x. Hence the two straight lines 



y= zb'*' form part of the locus of equation (6). 

 h. If 'llqn-^x l)e put for cc, / being an integer, the equation is un 



changed, whether q be odd or even. 



c. If q be odd the equation will be unchanged if lq7r-\-.i' be jnit for x. 



d. The curve repeats itself to the right and left, and also above and 



below, at intervals of qn \i q is odd., and at intervals of Iqn 

 if q is even. 



e. Straight lines parallel to yr=±./', and cu.tting the axes at intervals 



of qrr, or 2q7t, according as q is odd or even, belong to the 

 locus of equation (6). 



f. These straight lines divide the infinite plane of coordinates into 



equal squares for a given value of q. Each square contains a 

 similar and equal portion of the locus. If q is odd, that portion 

 is not always similarly placed, ibr it may have two positions 

 with respect to an axis. 



g. If q is even, isolated points at the centers of the squares {f) 



belong to the locus. 



h. The equation (6) is satisfied if sin ^=:0, and 9my=.0. Hence the 

 locus of (6) passes through the angular points of all the squares 

 formed by the lines sin a;z=0, and sin y^O (Art. V.) 



i. A few curves representing equation (6) are shown in figs. 40-4'?. 

 The axes are not drawn. Any point of intersection of straight 

 lines that is sui'rounded by an oval may he taken for the origin. 

 The several propositions of this article will be more easily un- 

 derstood by inspection of the curves. 



11, Equation (1) lolien a=l, bz=0, ni=:n:= /- . In this case the 



equation becomes, 



smy8iu±--y=:iiinx't^m^.x. (7) 



q' q 



The properties of the curves of equation (7) are in many respects like 



those of equation (tj). 



a. The two straight lines y:=: ±cc belong to the locus. 



b. If p and q are both odd, the equation is unchanged, if y or x is 



increased or diminished by multiples of qyt. 



c. If either /» or q is eve)t, the equation is unchanged if y or x is 



increased or diminished by multiples of 2q7r. 



d. The curve repeats in the direction of either axis ; at intervals of 



qTT if p and q are both odd, at intervals of '2q7r if either jd or q 

 is even. 



