THEORIA INTERPOLATIONIS METHODO NOVA TRACTATA. 
295 
cosi—cos« = 2sin4-(^—a) sin 4-(—t — a) 
cos t—cos b = 2 sin i (t—b) sin \ {—t—b) 
cos t — cos c = 2sinf(i— c)sin£(— t — c) 
cos t—cosd = 2 sin \ [t—d) sin\ (—t—d) 
etc, 
productum ex his factoribus fit, per lemma primum 
«ix . . — sino) w — sini|x(— t— a) 
Z X -f- 2jM _i X -f- 2(( t_i 
sin £ (x (t ■—a)sin£fx(—t—a) cosjxi — cosfxa ^ p 7^ 
2 /tt-a 2i“ -1 
Quum habeatur 
sin£p(i—d) — —sin^-p^—b) = sin^-[x(#—c) = —sin-|-p(i—d) etc. 
nec non cospa — cosp& = cos pc = cospi# etc.: manifestum est, productum 
in lemmate primo fieri etiam 
■ sin % [x (t h) 
ITI 
sin i (x (t 
(t — c) | sin*|x(< —d) , 
-1 I O 1 
-J— 2^“ 4 
nec non productum in lemmate secundo fieri etiam 
_ COS [xi — COS[x6 COSfxtf — COS[XC COS(x£ —COSjxd 
2 i “ _1 2 |it_1 2^"" 1 
20, 
Consideremus primo casum generalem art. 10, ubi X est formae 
a -f- a cos x -f- a"cos 2 x -f- a'"cos 3 x + • . • -f-cos mx 
—(— ' sin x —}— ,r sin 2 oc —t)sin 3 x -fi* . . . -f-1) sin mx 
sin a [x {t — a) 
atque p — 2jw + 1- Hic igitur erit 
sin+(i— i)sini(<—c)sm-Ki— d) ... = 2 „- sini(i _ g) 
= ¿¿¡(1+ 2cos(i—a) + 2cos2(<—a)-(-2cos3(£—o)+ ■ . • + 2cos»»(i «)) 
unde substituendo a pro t 
sin4-(<z — b)sm^[a — c)sin-|-(a — d) .. . = ^