320 
on the value of tan (sin 6) ~ sin (tan 6). 
and hence 
X + A'X s + B'X* + G'X 7 + D'X* 
- Y — AY Z - BY 5 - GY 7 - BY 9 
= 6 7 {3 A A' (A -A') + 2 (AS - A'B)} 
+ 6 9 {AA' (H 2 - A'*) + 6 AA' (B - B') + 4 (AC - A'G) + 10 (A*B - A'*E)}. 
Now let 
X — sin 9 — ^ 0 3 + jyo — nrio 9 7 + ...; A — — jr, B — C = — g^-g- 
F = tan0 = *0»+ & 0 7 --; A '= hZ'=&,c= & 
we have therefore 
AA' = -&, A-A' = -±, A+A'=±, B-B' = -l 
AB'-A'B = -jL, = 
Hence 
coeff. ^ = tV - iro> = so> 
coeff. d 9 = 2T(r + Y4 — YE + 3V’ = T5°6 5 
and the required equation is thus verified.