107

Hence z~c tang.p, and t~c sec. ?. More-ver by comparing the equal quantities L

and -jwe have y— ™ or by squaring eachside, and adding a? to each side of the result,we obtain (r 1 -}-c a ); or

since i' 2 -j -y 2 —£ 2 ~ s 2 = (s‘ 2 -|-c 2 ) or £ —

7 v/ z~ "f- c 2 ; hence by substituting this valueof £ in the equation ~ = ^ we obtain

\ C ~— 75 or by dividing by we have

\/z* + ci = t, and consequently c = v <* - z 3 ,and z = s/t'—c 1 .

Again from the equation j=~ we havex — ~, which, by substituting x/V+c 1 for t,

becomes x = — or by taking the cor-

W+c 1 , J

red fluent of each side, (observing that xand z are both = 0 at the same time) weobtain x = s/z % +c- —- c. Hence x -f- c— s/z'+c 2 , from which z = v/2 cx+x 2 , andby substituting t for x /z i +c i in the equationx -j- c = s /z 1 +c 2 , we have x -J- c = t orx—t - c—c sec.? - c=c (sec.® - 1) =---

V ‘ / cos. *.

4