98 Neioton, and Phillips on certain Transcendental Curves. 



in the figures. The origin is the point at the left of each figure 

 where the curve touches the axis of x. 



P p' 



It will be convenient at times to put m= — , and nz=z —, where/?, ^, 



p', and r/, are integers, and the fractions are reduced to their lowest 

 terms. 



4. Properties of the curves of Eq. (2). By inspection of the curves 

 on plates XIV and XV, and of their equations, we readily deduce the 

 following properties : 



a. The value of y is not greater than a. 



h. When either x or inx is a multiple of ;r, ;y=:0. 



c. There are maxima or minima values of // when wi tan ,r= — tan ma*. 



(/. When ni is rational the values of y repeat ; after qn if p and q 



are both odd ; after 'Iqn if either^:) or q is even. 

 e. When m is irrational the curve does not repeat its form. 

 /'. The curve is symmetrical about the axis of y, and about an axis 



through the middle point of each cycle. 

 g. If p or q is even., the curve is symmetrical about the point y=0, 



qn 



^' 

 h. There are, in each distance -Iq-n: along ,<■, p-\-q maxima, and an 



equal number of minima values of y. 



5. The value of y in Eq. (2) may be regarded as made up of two 



a a 



parts, since y=La&mx %\nnix=z~ go^[\ ~m)x — — cos(l-|-m)a.'. In 



fig. 22, where mz=L^, these parts are sej^arately shown. The continu- 



a 

 ous line represents the curve yz=: — qo^[\—^)j\ having one com- 



])lete oscillation in a distance of IQn along ii-. By laying ofl" below 



and above this curve the second part of y, that is— — cos(l -|-f)^, 

 we have the curve y=.a%mx sinf.«. 



6. Use of the auxiliary curves, Eq. (2). To draw the curves from 

 equation (1), even after all the usual devices for saA'ing labor have 

 been employed, requires the frequent solution of equations of the 

 form sin a; sin?;? 35:= c. This equation gives a set of values of x for 

 each cycle of the curve. To find each value of x requires a solution 

 l)y trial and error, a very simple process, but when often repeated 

 quite tedious. By the curves figs. 1-37 carefully traced on cross- 

 section paper we may by merely running the eye along the line y=.c 



