ON THE USE OF LOGAKITHMIC COORDINATES. 177 



the curves in values for / which are equal, each to each, to the inter- 

 sections on the first straight line. 



Thus on the diagram the figure Aa' is a rectangle whose sides AA', 

 aa' are bisected by the axis of %(,, and the lines «A, 6B, cC, a' A', h'W, 

 c'C are all equal and parallel. 



This property would be of great value in the origination or verification 

 of tables of these functions. 



If D be the intersection of the sinh and coth curves, it follows that 

 'D'D'd'd is a rectangle whose sides are parallel and perpendicular to the 



axis of u. 



If Q be the intersection of sinh to and cosech m, Q must lie on the 

 axis of u. It follows at once that EQE' is a straight line perpendicular 

 to the axis of n. 



51. For the point E we have 



whence e"=l + N/2, 



e-"=N/"2-l. 

 Thus when u has the value given by 



MM = l0g 10 (1 + V 2) 



(m= -88 approximately), 



sinh ?6= cosech 'M=1 

 cosh M=coth i(,^=\/'2i 



sech M=tanh ?4=^— -. 



The lines «=2 and u=\ are inserted in fig. 6. RR' is divided into 

 four equal parts by E, Q, E'. 



52. The value of u which makes cosh ^^=cosech u is given by 



sinh 2m =2, 



whence e2"=V5 + 2, 



e-"-»=v'5— 2. 



The value of « is given by 



2MM=logio(^/5 + 2), 



and is approximately 0*72. 



The value of cosh u is given by 



2 cosh^ M=cosh 2m + 1 

 = v/'5 + l. 



Thus 



cosech 16= cosh u 



sinh M=sech u 

 tanh M= 



1898. N 



