134 Mr. Ivory on the figure requisite to maintain the equilibrium 
And again, 
a x -f- by c % = R r y = R' r' y ' ; 
y = (JO \/ 1 — p. y/ 1 — [a / 2 Cos. (js — ns ' ) 
y' = pp' y/ 1 — p 2 . V 1 — p 12 Cos. ( q — q'). 
Wherefore, 
R 2 — 2 R r y 4- r* = R /2 — 2 R' r' y -f- r'\ 
Farther, from the assumed equations we readily derive these 
values, viz. 
Tan. zs'=Y‘jri' Tan. q' t 
/ ' 
v- 
Vi — 
A' A" 
M ’ 
1 — j / 1 
M = Vk ' 2 h " 2 Sin. 2 q'+ h '* Cos.’ 9' 
r vT— = II' vr 
l/2 
M 
• A' A" ' 
And, if we now take the fluxions of the first and second of 
these formulae ; observing, in the second operation, to make 
q\ and consequently M, constant ; we shall get, 
d zs' = dq 1 . 
k' h" . h‘ h" 
d pf dp' 
(I — !*/»)*““ (I —pr*)i 
k A' h' 
AM 
R3( 1 _ (J# ' 2 )2 == R'3( 1 _y.) 
f w 
A' 3 A" 3 
And, by combining these formulae, we obtain. 
But, in § 3 it has been shown, that, 
vw\ rr 
] „3 __ // R 3 dp.' dn' 
J JJ V/I‘-2R 
d. 
d r 
and hence, by substitution, 
-f- r a 
f d .m\ rr 
f rI J J.3 * k" // R' 3 dp' d q' 
\ dr / A A' A" JJ v / 1< 2 R'r' y ' + r' 1 ‘ 
