the Hyperbolic Trigonometrical Functions of 6. 239 



d shin 6 



Also DB = Sa = 6.S shin 0=6. 



and ED = 8c = b.8co*\i6 = b. 



dd ' 

 d cosh 6 

 ~~dd 



86. =b cosh 6 .86; 

 86 =b thin 6. 86. 



And the triangle BED being similar to the triangle A B, 



BE _ DB = b . cosh 6 

 ~b ™ c c 



2(BE 



86=86. 



= 6. 



Therefore 6 can be obtained by summing such small 

 elements as BE, and dividing the result by b. BE, in fact, 

 is that component of the motion of B which is at right angles 

 to AB. 



Fio- 2. 



/fc/sz/vc £dg£ 



Suppose that at the point A there is a circular pin in a 

 drawing-board, but capable of moving round a vertical axis 

 exactly coinciding with the geometrical point A. 



In the head of this pin is a horizontal slot in which a bar 

 slides. 



This bar is enlarged at one end, as shown in the sketch, 

 so as to carry a small wheel moved, by friction against the 

 board, about an axis exactly coinciding with the centre line 

 of the bar. 



It will be clear that the movement in arc of a point in the 

 circumference of this wheel will give the quantity 2 (BE) or 

 6 . b if the point of contact of the wheel with the board is 

 carried along C B. If the circumference of the wheel is 

 equal to b, the turns of the wheel will give 0, and these may 

 be given by arrangements (not shown in the sketch) similar 

 to those in Amsler's planimeter. 



