OF MULTIPLE: ARCS. 289: 
§7. In like manner, we may find sin m 0, m being a positive in- 
teger. In (6) substitute for ¢” its value in (8). Then 
(m—3) | (m—4) 54 
eer! 
2° 2k-* sin m O=t-"—(m—2) "B+ (m— ea (m—4) m4) _ Bo. 
2h-1 4" sin m 06=1—(m—2)é? + ——— — &e. 
sin 6 
But because 2¢ cos 0=1, and 24 sn 0=h, k= saad 
Therefore 
oe n m 6=(2 cos 6)" —(m—2) (2 cos 6)"-2 + &e. 
, sin m O= sin 6§ (2 cos 0)"-1—(m—2) (2 cos 6)"-3 + &e.}. 
§8. Another very simple instance of the application of the formuls 
which we have obtained is the following. By (2), 
TTT, =T,—?(T,_»—02T,_) = TOT, FET 
=T,—#T,_, +¢*T,_,—¢°T,_, +... +(—) 4? Te 
+(—1)°+2 2204UT, oe) 
Substitute for T,.,, T,, &c., their values in (7), and divide by 2¢”. 
‘Then 
cos ~ 6—cos (n—2) 6+ cos (nx—4) 6—...... + (—1)° cos(n—2c) 6 
=) cos (n+1)9—(—1)°*cos(n—2ce—1)0 ' 
__ cos(n+ 1)6—(—)'+e0s(n—2e—1)8. 
2 cos 0 
In like manner, 
sin 2 6—sin(n—2)0 + sin(n—4)0—....2. +(—1)*sin (n—2c) 6 
sin (n+ 1)0—(—1)°*? sin (n—2ce—1)6 
v 2 cos 0 
Vou. VIII. NN ee 
