842] 
319 
842. 
ON THE VALUE OF tan (sin 0) — sin (tan 0). 
[From the Messenger of Mathematics, vol. xiv. (1885), pp. 191, 192.] 
The following equation is given p. 59 of the Lady’s and Gentlemans Diary for 1853 : 
tan (sin 6) — sin tan 6 = + &c. 
Write in general 
X = 0 + A0 3 + B0'° + C6 7 + D0 9 + 
Y=0 + A'0* + B'0* + C'fr + D'0 9 + .... 
Then, as far as 0 t] , we have 
X + A'X 3 + B'X 5 + C'XJ + D'X 9 
0 + A 0 s + B0 5 + 
CP + 
D0 9 
+ A' {0 s + 3A0 S + 3 (A- + B) & + (A 3 + 6AB + 30) &} 
+ B'{ 6> 5 + 
5A0 7 + 
(10A 2 + 55) 0 9 } 
+ 0' { 
0 V + 
7 A 0 s } 
+ D'{ 
0 9 } 
+ 0 
+ 0* (A + A') 
+ 0fB + 3AA' + B r ) 
+ 0 7 (C + 3 A 2 A' + 3 A'B + 5AB' + O') 
+ 0 9 (D +A 3 A' + 6AA'B + SA'C+10 A*B' + 5BB'+ t 7AC + D'),