414 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
The elements of these transformations are given at page 358, namely, 
2sinV=l4-? sin^-v^— cos-v//\/r„ and sin^/. 
From this last equation we derive (1 — w sin^<p)(l — m sin*^) = I(l — m^sin^-v//). 
Now as <!> shall have ^/mM2n-2nmsmy-l ^ 
l-7ism2<p " 2\/mn L J 
O .,- f 1 Vm.&m’h \2n—nm — wm, sin^\I/ -1- mn cosvp V 1,1 
r putting for sin a its value, w<P„= ‘ — t! 
In the same manner, we may find 
. m, sin4' r 2m — mn — mni, sin^4' + cos\J/ v 1 1 
7n(P„,= 
(421.) 
(422.) 
(423.) 
2Vmn [1— ?w,sin®4>] 
Adding those equations together, and recollecting that we shall get 
, ~ ^ m,i^ sinrj/ . ^ m,'^ mn cos\}/ sirn}/ ^1, / in a \ 
' 
Now as ( 1 +j) ( 1 —j)i and \/ mn= 
- (n<l>.+mO.) = - ( 1 -j) sin4-- (1 
In (186.) we found 
2Jd9^/I= ( 1 ( 1 -J) sin^^/ (426.) 
Adding this expression to the preceding, the terms involving sin-^/ will disappear. 
We must now compute the sum of the coefficients of 
Since this coefficient becomes ^^^[^+*;^-2(l+i)]- 
Or as m-\-nz=i^-\-mn, this coefficient may be written [^+^"^~2(1 4-j)J 
2 “1 
Or as mw=m^(I +j)^, it becomes finally, — I (427-) 
Hence 
S+;-«.+i)]{i#).f^rraI|-'lf^i; 1“’ 
d^f/ 
— d<p (1— n)F dip n {n—m){n—m) 1 F d\{/ /inn\ 
And {n—m) [(^ ~~V^ ~ V^n (l+i)J[l-m, 
!M 
Now as n-\-m—i^ — mn, {n-\-my=i* — 27nn'f . 
Hence {n — »2)^=i^+2 mni^ + nfn^ — Amn, 
and as i^=z{\-\-jy{\—jY, substituting 
{n—mY=z(\ +jT( 1 -J)"+ 2wX 1 -YjYi 1 -i) 1 +7)' “ ^ +i)*’ 
