PROFESSOL* a. H. DARWIX OX ELLTPSr)TDAL TIAR.NrOXrC ANALVSIS. 
527 
Now for p/ associated witli F^, 
7.3 — lA [ 1 “ (*' +)] ^ 
<h 
_ i__. 
7 () 8 ' 
and for P,’(!r/, associated witF Fo, 
'rherefore 
73 
' — ]"6 [ 1 K ^ 1)] ’ 75' — 7 18 • 
2__ <- + i: 
2;+] i-1 ! 
i±U 
Yi + l f- 1 ! 
4-^0 + 2)(?-f 3)(i— !)(?. — ^) — (^' ~ 1 )(^-“2)? ('i'.-|-l)] ], 
P (i - 1) (i - 2) + xis (^' -1) (^’ - 2) 0'“ 3) (^ - 4) 
In the present case we cannot use 2 as an aliridgement, since it is infinite ; I 
therefore now write 
Fffectino- the reductions we liave 
2 z + 1! 
f 2 ; 
+ 1 i-1! 
j — 2(2?d-l)+4)d- ~ 4/' + -:VJ + 30—10 (2 {+ 1)] j-, 
+ Tk/3TW+---|-; + 10-l(5(2/+l)(,/+5)]f. 
'fhe former of these is associated with the latter with 
In the cosine aiul sine functions we have 
p^_o — p_i = 0, p,_ I = />_:■ = 0, and 
(^,1)^ = X “ 4 cos2fj6 + /3(p.^cos 2(;^) — pocos4(^) 
+ /^'[^(Ps)' + 7 pC”« 4 c/> - {p- + h iihf) cos ChI >1 
( (Cf)’ = X + ^ cos 2f/) + (3 ( p, cos 2c/) + 2h cos 4c/)) 
+ + p:, cos 4c/j + (p- d- COS 6^]. 
As far as material, we then have 
1^/)' 
= o 1 c — -r ^ - T- TiF-z T -] 
+ (- ^^ + + /3p3) cos 2(/), 
+ ^'[(/h)' + hP?, + TnllH*- — (1 — 2/3p.3 — j/8) cos 2c/)j-; 
(1 — /Seos 2</)) 
f = o {1 — -4-yS + / 3 '[(po)- + ypPo + -]+]} 
_ I ' 
— N 1 
