TRANSACTIONS OF SECTION A. 44] 
fsin’r.Fdx, foos’xr.Fdx,  ftanra. Fax, 
and for some others of similar forms; also some that contain the denominator A, A®, 
A*. He did the same for another class that had the Z instead of the F. 
For each of these formule he gave the initial integrals for p=1, 2, 3, ete. 
From these integrals, again, he deduced some other general ones with the factors 
EF”, E*, EF; also with the initial cases. 
In obtaining these reductions it was for some initial values necessary to admit 
certain new transcendents. 
Now from the last-mentioned integrals, it is sometimes possible to eliminate 
these transcendents, and so obtain some new results, viz.: 
[7 cos Dep e_ =22 tan jv- = S/S ee 
sin? 2 sin 2’ sin* x 
\ [F cos 2x.dx = FS (AF—.2) +sin x cos v. F?, 
[P og Bp 8 A ae tan 6 D se) } Soma 
cos?x 1—k* (cose COS X 
E? cos 22.dx = plliez” — 3(1 —k*) — 2a? ]2? sin x cos x—8A°E 
—(8—8h? + 3h')a}, 
[EF cos Qa.dx = a {(A2F'+ 8E)a — (2-22) 2u—(2-8EF)E sin x cos.2}. 
In order to obtain the general reduction-formulz for these last results, it is neces- 
sary to put p+2 for p in the theorems Ixxiii., lxxv., Ixxy., Ixxvii.; to multiply 
with 2, and take the difference between this result and the original one. Then by 
consecutively substituting the relation 2 sin’+? x=sin” «(1—cos 2x), it follows 
that 
[sim Px cos 2x. F*dx = : = [2a +k?) |sine-2 & cos 2u.F*dx 
2)2k? J 
(p+1) (p+ 
— {2(p=1) (p=2)—@p-1)k"} |sinPtx eos 2x.F°de 
+ {4(p—2) - @p— 1p} [sine Fede 
+4 [sine-*x cos 2a.dv—4 sinP-22 cos 22.Ab 
— {(1—pA*) cos 2x—(1 + 2 sin? x)A?}2 sin?-3x cos oF | Pay fl 
fsinre cos 2v.E Fav = oe [ p+ (p+ 2)k242p [sine cos 22. EFde 
— {2(p—1) ( p—2)+ (Qp— 14°} {sin »~4y cos 2u.EFdx 
+ {4(p—2)—(2 p—1)k%} | sin’, BFde 
+ 4|sin ey cos 2x. A*dx + 4x*(sin Px cos 22' cos wae 
— (E+ A°F’) 2 sin?-2x cos 22.A 
—[(8—pA?) cos 2x —(1—6 sin? x)A?] 2 sin?-8x cos 2. EF | > ule 
