Equations occurring in Wireless Telegraphy . 209 



meet OX in the point N 1? and let the perpendicular through 

 R x on OX in its new position meet OX in the point A 1# 



As before, let a pin-prick be made through N : into the 

 lower paper and let the point thus marked on the lower 

 paper be denoted by N x , and let a pin be stuck through A T 

 into the drawing-board and let the point thus marked on 

 the lower paper be denoted by A x . 



The pin through A is now to be removed and the squared 

 paper turned through a very small angle about the pin 

 through A}. 



Points N 2 and A 2 are now determined in a similar manner 

 as were N x and A l5 a pin-prick made through N 2 into the 

 lower paper and a pin stuck through A 2 into the drawing- 

 board. 



The points on the lower paper corresponding to N 2 and A 2 

 may now be denoted by N 2 and A 2 respectively, and the 

 process may be repeated an indefinite number of times. 



We thus get two sets of pin-pricks in the lower paper : — 

 A t , A l7 A 2 , .... A n , and No, N l5 N 2 , . . . . N» ; and the two 

 broken lines A^Ao . . . . A n and N N 1 N 3 . . . . N„ approxi- 

 mate 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 the A curve and the N curve respec- 

 tively. 



Now t , provided that the curvature of the A curve remains 

 of one sign in a given stretch, it is evident that 



A A 1 + A 1 A 2 + .... +A, l _ 1 A n =A A 1 + A 1 A 2 + .... 



4- A„ _ i A „ = A A n . 



If, however, in one part the curvature of the A curve is 

 positive and in another negative, lengths must be taken with 

 opposite signs along these two parts if this equation is to 

 hold *. 



With this understanding, we see that A A ?i tends in the 

 limit to the length of the A curve between the corresponding 

 points. 



Again, PN , PN l5 PN 2 .... PN ;l are the perpendiculars 

 from P on the various positions of the line OX, which in the 

 limit becomes the tangent to the A curve ; and so PN , PN lr 

 &c. become successive values of p. 



* It is easy to see that a maximum or minimum in x as a function 

 of y will correspond to a point on the A curve where the curvature 

 changes sign. 



Phil. Mag. 8. 6. Vol. 43. No. 253. Jan. 1922. P 



