Equations occurring in Wireless Telegraphy. 21o 



Let this line intersect the curve DE in the point B^ 

 which will be very near to E . 



Let the perpendicular through P on OX in its new position 

 intersect OX in N x and the curve (5) in T 1? and let the 

 perpendicular through B^ on OX in its new position inter- 

 sect OX in Mi. 



Let a length M^Ax equal to PT X (or i\) be measured off 

 along the axis of x as before and let a pin be stuck through 

 Aj into the drawing-board and the pin through A be 

 removed. 



Let the point of the lower paper marked by the pin 

 through A 1 be denoted by A x . 



Also let a pin-prick be made through N 2 into the lower 

 paper and let the point of it thus marked be denoted by JSx- 



Let the process, as described, be repeated an indefinite 

 number of times and we obtain two sets of pin-pricks in the 

 lower paper, A , A l} A 2 , . . . . A„ and N , S" 1? N 2 , .... N», 

 and these approximate indefinitely to two curved lines as we 

 take the small angles of rotation of the squared paper more 

 and more minute. We shall call these two curved lines the 

 A curve and the N curve respectively. 



It is evident that 



A A 1 + A 1 A 2 + +A„_ 1 A*=A A 1 + AiA 2 + .... 



and, if the A curve has its curvature of one sign in the 

 interval, then this = A A„, so that A A ?l tends in the limit to 

 the length of the stretch of the A curve. 



Also PN , PNi, PX 2 , . . * . PN„ are the perpendiculars 

 from P on the various positions of the line OX w T hich in the 

 limit becomes the tangent to the A curve ; so that P-N > 

 PN 1? PN 2 , .... PN n become successive values of p for this 

 curve. 



Until we have shown that the A curve is the one required 

 we shall distinguish the p and s of this curve by accents. 



Now we have 



and this is the ordinate of the curve (4) corresponding to 

 the point M„ on the axis of #, while 



AoA»+A B M n =A M„ 



= OM„-OA , 



or A Q A n + r Q = x — OA . 



