ON THE CALCULATION OF MATHEMATICAL TABLES. 



dn^(l-/)G dn^(l+/)G 



71 



\/y' 



>/y' 



ka=b^y'tnl (1-/)G, A'ft' = 6s/ytnJ(l+/)G, 

 dn^(l-f)G /i + «gn;^K , ,,1., ...^ /1-sn/K 



PQ = (rt - 6)sn2/L, FJ = (« - 6)cn 2/L, FI = (a + 5)dn 2/L, and 

 OJP=am 2/L, and so on, showing the geometrical interpretation of 

 the elliptic function and its Quadric Transformations. 



But the A, B, C, D functions cannot be shown in the figure ; and 

 E(r), F{r) arise in the rectification of the elliptic arc. 



In the motion of the simple pendulum, oscillating through a finite 

 angle, four times the modular angle, the pendulum beats the elliptic 



function of the time t, such that t=^fT, if T is the beat in seconds. 



The lemniscate function is required when the pendulum swings 

 through two right angles. 

 From the relation 



1- .^f._,\/l-sn/K or^^^'^^^^ + i^OG , .,_ 1..,,^ 

 -^^^- ^ fWK ^nT(45)G =M45-2<^(rK)], 



^y'tn^(l-f)G: 



the column of Legendre's <^ in Table III can be deduced from that in 

 Table II ; and so also in II from I. 



A slide rule may be used in a first approximation to the nearest 

 degree, to read off from the scale of tangents on a fixed setting of the 

 cursor. 



K 2 



