DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 409 
1 j 
We have also and hence 
= (403.) 
and therefore 
It is worthy of especial remark that this coefficient ofJ ^^^ j is precisely the 
in form as the coefficient 
same 
In vv 
The preceding equation (402.) may now be written, 
o / — r ^ 
■[ 
^ mn siiKf coS(p'i 
Vl 
J- 
(405.) 
Let n^, be analogous quantities for the derived spherical ellipse 
+ / C d^/ Cd\{/ , r -v/m-n, sin^I/ cos\|/n 
p 
Let q, q,, q,„ &c. denote &c., and put r, r^, r,,, &c. for 
V mn Vmin^ Vm^in^^ 
(1 +j ) ( 1 +Ji), ( 1 +i) ( 1 +j,) ( 1 +iJ, ( 1 +j) ( 1 +i;) ( 1 +jy ( ] +jj, &C. Let also 
n, T, Tp T,,, &c. denote the arcs, whose tangents are 
a / mn sin^ cos<p sin4/ cosvl/ sin^/^ cos\J/, 
A/l-i^sin^^’ v"i_ 22 sin 24 /, 
Making these substitutions, and writing Q, Q^, &c. for the coefficients of ’ 
Taking the derivatives of these expressions, we may write 
Subtract (a,.) from (a.), (b^.) from (b.), and (c,.) from (c.), the integrals of the third 
order disappear, and we shall have 
2cr-cr^={q,r-q)^^+-^-D, 
2(r g^r)J^ + T,- T 
^ III *’’//;/ (? infill ~ ^ Ilf ii)^ ^ J "i" III II 
3 G 2 
(a;.) 
(b.) 
( 407 .) 
