Mr. A. Cay ley on a Transcendent Equation. 21 



And in like manner 



singdu = — ( e «#*«— «-*>**) 



1 /l + taiigW? ] — tanjwt\ 

 2i \1 — - tan hui 1 + tan hid) 



+ 

 2 tan | wz sin we 



z(l — tan 2 |wz) z'eoswi 

 or, as these equations may also be written, 



e u + e~ u 



sec gdu = cos ui 



2 



gW g — U 



tan gdu = -sinwi = - — • 



i £ 



And from these equations we have 



secgd(u + v) = sec gdu . secgdv + tan gdu . tangdv, 

 tangd(u-\-v) = tan gdu . secgdv -f tangdv . secgdv, 



or, what is the same thing, 



7/ x sin qdu+ sin qdv 



sin qd(u-\-v) = , . —5 ^ — =— , 



^ 1 + sin y# ?« . sin gd v 



7/ . cos qd u . cos q d v 



cos qdiu + v) = -, ? — ,- * — >— : 



17 1 + sm gdu . sin gdv 



which form is at once obtainable from the formula) 



. . sin am u cos am v A am v + sin am v cos am u A am u 



sin am[u + v) = z /2 . a 2 , 



v 1 — at snr «w u sur awt t; 



, x cosamucosamv— sin am u sin am vAamuAamv 



cos am{u + v) = = 1-5 5- :— 5 



v ' 1 — Arsin^ amusm^amv 



. . N AamuAamv — # 2 sin amu sin «????; cos m« cos amv 

 l\am{u + v) = = ya . a r-« , 



observing that for /c = l we have Aarn= cos am, and that the 

 numerators and denominator each of them divide by 



1 — sin am u sin am v. 



2 Stone Buildings, W.C., 

 May 7, 1862. 



