422 
ME. C. W. MEEEiriELD ON A NEW METHOD OE APPEOXIMATION 
of a square form. For the common form of the elliptic radical \/(l— sin^ d . sin^ p), our 
choice is practically limited to 
(1) r=l, (2) r=cos (p, (3) r=sm0.cosp, 
(4) r=cos 6, (5) 7’=cos 6 . sin p. 
And on these suppositions I now proceed to the integration of the general form of the 
reduced approximant for J^(l — p)~^dp=^zdp. I omit mention of the con- 
stants of integration, because very slight changes in the function may alter them. The 
lirst of our three cases require, as they stand, no constant, and these are the most useful 
cases. 
(1) r=l, — N=sin^ sin^ 
2 
1 — sin^ S . cos^ — . sin^ 9 
= 1 — sin^ ^ . cos^ tan"'|^l — sin® ^ . cos^ tan®|- 
(2) 7'=cosp, r®— N=— cos® sin®<p, 
2 cos ip 
2 cos f 
cos® p + cos^ Q . cos® ^ . sin® p ^ — cos® 5 . cos® . sin® p 
^^zdp= ( 1 — cos® 6 . cos® logj 
. -f- sin ip^l —cos® 5 . cos® 
. — sin 1 — cos® fl . cos® 
. 
(3) r=sin^. cos(p, — N= — cos® 
2 sin 5 . cos p 
— COS® . sin® — sin'-" 0 . sin'-' p 
|2 A / 
^ 7 . _i 111 —cos® 9 . sin® -l-sin 9 . sin p 
(l-cos’ 0 , sin’ ^)-* log ) _ 
( 1 — cos® 9 . sin® —sin 9 . sin 9 
> . 
(4) 7'=cosd, r®— N= — sin® ^ cos® 
2 1 
cos 9, . o„ o^Tr o’ 
1 + tan® 9 . cos® — . cos® p 
— sin® ^ . sin® tan ‘jcos ^ . tan — sin® ^ . sin® |. 
(5) r— cos^sinip, r®— N:r= — cos® (p, 
2 cos 9 . sin a 
^ — r 
cos® 9 . sin® (p -fcos® — . cos® p 
