extensive Class of Spheroids. 
55 
By means of the last formula the fluent in question will be 
reduced to the integration of expresssions of this kind, viz. 
1 . (1 — . d\b ; a research with which mathematicians 
are familiar. In the first place when s is even ; then, consi- 
dering the definite fluent between the limits y = — 1 and y=i ; 
we have 
1 * (l — f 1 ) 1 • dp = o : 
and indeed, supposing P to be any odd function of y, we have 
more generally ^P . dp = o, between the same limits. In 
the second place when s is odd ; then, taking the definite 
fluent as before, 
I* ' ('-f* ) • d >*= T ■•*+,• • ifcit - 
2 
2 + S 
\ r 1 r s ' z i+s * zi+s —2 
The observations that have already been made are sufficient 
to point out in what manner the expressions of the fluents 
under consideration may be formed with great practical com- 
modiousness. 
5. Let y, y, y denote the cosines of the three sides of a 
spherical triangle ; and let <p be the angle opposite to the side 
whose cosine is y : then, according to what is taught in sphe- 
rical trigonometry, 
<y = + V 1 — y' • V 1 — \f* . cos. <p : 
suppose farther that/ = j r — 2 ra . y -|- a* J T , 
and let 
7 = Q ' 01 • i + Q {,) • F+ Q w ■ £ + Q [>) ■ &c. 
/ ~ r • ~ - r- • ~ r* * ^ * r *+l 
it is required to expand which is the same function of 7 
that C (,) is of (ju, into a series of the cosines of q> and its mul- 
tiples.* 
• Mec. Cel. Liv. 3e, No. 15. 
