121

= 56° 28' = the angle ABC which was tobe found.

16. Corollary. If in the expressionm log. tang. (45°+^) = cosec. ? =obtained in the last problem, we substitute

for log. tang. (45°-fi <p) according to Case

IX. of art. 9, and for ~— we substitute

sin. <p

1 .4- ~ = — according to Case X. of the

same article, we have ~ = — or y = — for

a particular relation between «/, c, t, and z,which obtains only when i is a minimumwith respect to y.

17. Problem. Wherein it is proposedto find the subtangent to the catenary.

From article 8, we have ?/ = — or 4 =—

u Z y C

whence, by multiplying each side by y,= the subtangent required.

18. Corollary. By putting s for thesub tangent, we have s — ~ • and from ar-ticle 16. when t is a minimum with respect