IX FOCAL PLANE OF TELESCOPE. 337 



straight lines, one part of the curve will lie to the right and the other 

 to the left of the line ; the points of intersection of the line with the 

 curve will, for large values of x, he nearly midway between the two 

 points and nearly coincident with the inflexion points of the second 

 set. 



Represented graphically, the curve shows a succession of steps, at 

 nearly equ;d horizontal intervals ; the height of the consecutive steps 

 becomes smaller and the rate of decrease diminishes, as we recede 

 further from the axis of y. 



Fig. 1. PI. XVI. shows the curve from «=0 to œ = 10 ; for larger 

 values of a*, the curve is given in Fig. 2. The positions of the inflex- 

 ion points of the 1st set are given by the intersection of the two dotted 

 lines, of which the one represents the inflexional tangents and the 

 other the abscissae of the inflexion points. 



It will be further shown that the mean curve is approximately 

 ;i rectangular hyperbola. 



k 2 



§ 3. Expressions for J?(x)+Jftx) 



Different expressions can be found for y. For small values of x, 

 we m;iy expand it in powers of .r, by means of well-known definite 

 integrals 



Jq\x) = -^—JJ ('2x sin(o)d(o , 

 J\i x )~ —\Jé$ >x sintb)cos 2m dco 



2 r n 

 J Kx)+Jf(x)=:—jJa 2 (2x sinco)cos^co dco ; 



