88 
ON A TRANSCENDENT EQUATION. 
[320 
and in like manner 
sin gd u = ^ i (e'" du — e~ l z Au ) 
1 /1 4 tan Am 1 — tan kui\ 
2i \1 — tan \ui 1 4 tan \uii 
2 tan \ui sin ui 
i (1 — tan 2 \ui) i cos ui ’ 
or, as these equations may also be written, 
sec gd u = cos ui = ^ (e v + e~ H ), 
tan gd u = - sin ui = (e" — e~ u ); 
and from these equations we have 
sec gd (u + v) = sec gd u . sec gd v + tan gd u . tan gd v, 
tan gd (u + v) = tan gd u. sec gd v + tan gd v . sec gd v ; 
or, what is the same thing, 
sin gd u 4 sin gd v 
sin gd (u 4 v) = 
cos gd (u 4v) = 
1 4 sin gd u . sin gd v ’ 
cos gd u . cos gd v 
1 4 sin gd u. sin gd v ’ 
which forms are at once obtainable from the formulae 
sin am u cos am v A am v 4 sin am v cos am u A am u 
3!